No, Rolle's Theorem cannot be applied to the function [tex]f(x) = (x - 1)(x - 5)(x - 6)\\[/tex] on the closed interval (4, 6].
Rolle's Theorem states that for a function to satisfy the conditions of the theorem, it must be continuous on the closed interval [a, b] and differentiable on the open interval (a, b). Additionally, the function must have equal values at the endpoints of the interval.
In this case, the function [tex]f(x) = (x - 1)(x - 5)(x - 6)[/tex] is continuous on the closed interval (4, 6], as it is a polynomial function and polynomials are continuous everywhere. However, the function is not differentiable at x = 5 because it has a point of non-differentiability (a vertical tangent) at x = 5.
Since f(x) fails to meet the condition of differentiability on the open interval (4, 6), Rolle's Theorem cannot be applied to this function on the interval (4, 6].
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1. Which of the following is a vector parallel to (5,3, -1)? A. (5,3,1) B. (15,-9, 3) C. (50, 30, 10) D. (-10,-6, 2)
The vector (5, 3, -1) is parallel to the vector (50, 30, 10).
To determine if a vector is parallel to another vector, we compare their direction. Two vectors are parallel if they have the same direction or are in the opposite direction. We can achieve this by scaling one vector to match the other.
In this case, we can see that the vector (50, 30, 10) is a scaled version of the vector (5, 3, -1). By multiplying the vector (5, 3, -1) by 10, we obtain the vector (50, 30, 10).
Since both vectors have the same direction, they are parallel. Therefore, the vector (50, 30, 10) is parallel to the vector (5, 3, -1).
Among the given options, the vector (50, 30, 10) corresponds to choice C. So, option C, (50, 30, 10), is the correct answer as it is parallel to the vector (5, 3, -1).
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(5) Consider the hallowed-out ball a' < x2 + y2 + x2 < b>, where () < a < b are con- stants. Let S be the union of the two surfaces of this ball, where the outer surface is given an outward orientation and the inner surface is given an inward orientation. Let r=(c,y,z) and r=|r|. a) Find the flux through S of F=r (b) Find the flux through S of F = r/r3
(a) The flux through the union of the two surfaces of the hallowed-out ball of the vector field F = r can be found using the divergence theorem.
(b) The flux through the same surfaces of the vector field F = r / [tex]r^{3}[/tex]can also be calculated using the divergence theorem.
(a) To find the flux through the union of the outer and inner surfaces of the hallowed-out ball of the vector field F = r, we can use the divergence theorem. The divergence theorem states that the flux of a vector field through a closed surface is equal to the triple integral of the divergence of the vector field over the volume enclosed by the surface. Since the ball is hallowed-out, the enclosed volume is the difference between the volume of the outer ball (b) and the volume of the inner ball (a). The divergence of the vector field F = r is equal to 3. Thus, the flux through S of F = r is equal to the triple integral of 3 over the volume enclosed by the surfaces.
(b) Similarly, to find the flux through the same surfaces of the vector field F = r / [tex]r^{3}[/tex], we can again apply the divergence theorem. The divergence of the vector field F = r / [tex]r^{3}[/tex] is equal to 0, as it can be calculated as the sum of the derivatives of the components of F with respect to their corresponding variables, which results in 0. Therefore, the flux through S of F = r / [tex]r^{3}[/tex] is also equal to 0.
In summary, the flux through the union of the outer and inner surfaces of the hallowed-out ball for the vector field F = r can be calculated using the divergence theorem, while the flux for the vector field F = r / [tex]r^{3}[/tex] is equal to 0.
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In questions 1-3, find the area bounded by the graphs - show work thru integration. - y= 4 – x2 y = 2x – 4 on (-1,2)
The area bounded by the graphs of y = 4 - x^2 and y = 2x - 4 on the interval (-1,2) can be found using integration.
To find the area bounded by the two given graphs, we need to determine the points of intersection first. Setting the equations equal to each other, we have:
4 - x^2 = 2x - 4
Rearranging the equation, we get:
x^2 + 2x - 8 = 0
Factoring the quadratic equation, we have:
(x + 4)(x - 2) = 0
This gives us two possible x-values: x = -4 and x = 2.
Next, we integrate the difference of the two functions between these x-values to find the area between the curves.
∫[a,b] (f(x) - g(x)) dx
Applying this formula, we integrate (4 - x^2) - (2x - 4) with respect to x from -1 to 2:
∫[-1,2] (4 - x^2) - (2x - 4) dx
Simplifying the integral, we get:
∫[-1,2] (8 - x^2 - 2x) dx
Evaluating this integral, we find the area between the curves:
[8x - (x^3/3) - x^2] evaluated from -1 to 2
After calculating the values, the area bounded by the graphs is determined.
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(This is one question, please answer all the sub
points!!!! I will give a thumbs up I promise. Have a great
day.)
f(x) = 2x² in(x), x > 0. fa = x . (A) List all critical numbers of f. If there are no critical numbers, enter 'NONE'. Critical numbers = (B) Use interval notation to indicate where f(x) is decreasi
a. The critical number of f(x) is x = e^(-1) or approximately 0.368.
b. The intervals of decreasing and increasing values of f(x) using interval notation:
f(x) is decreasing on the interval (0, e^(-1))f(x) is increasing on the interval (e^(-1), ∞)A) To find the critical numbers of f(x), we need to determine where the derivative of f(x) is equal to zero or undefined. Let's find the derivative of f(x) first:
f(x) = 2x² ln(x)
Using the product rule, we have:
f'(x) = 2x² (1/x) + ln(x) (2x)
= 2x + 2x ln(x)
To find the critical numbers, we set f'(x) = 0 and solve for x:
2x + 2x ln(x) = 0
Since x > 0, we can divide both sides by 2x to simplify the equation:
1 + ln(x) = 0
ln(x) = -1
Taking the exponential of both sides, we have:
x = e^(-1)
Therefore, the critical number of f(x) is x = e^(-1) or approximately 0.368.
B) To determine where f(x) is decreasing, we need to analyze the sign of the derivative f'(x) in different intervals. Let's consider the intervals (0, e^(-1)) and (e^(-1), ∞).
In the interval (0, e^(-1)), f'(x) = 2x + 2x ln(x) < 0 because both terms are negative. Therefore, f(x) is decreasing on this interval.
In the interval (e^(-1), ∞), f'(x) = 2x + 2x ln(x) > 0 because both terms are positive. Thus, f(x) is increasing on this interval.
Therefore, we can represent the intervals of decreasing and increasing values of f(x) using interval notation:
f(x) is decreasing on the interval (0, e^(-1))
f(x) is increasing on the interval (e^(-1), ∞)
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clear legible work please
4 Find the integral of S 1 dx when n=10 In x 2 a) Solve using trapezoidal rule b) Solve using midpoint rule c) Solve using simpons rule State approximate decimal answers
the integral of 1 dx when n = 10 using different numerical integration methods, let's use the trapezoidal rule, midpoint rule, and Simpson's rule.
a) Trapezoidal Rule:The trapezoidal rule approximates the integral by approximating the area under the curve as a trapezoid.
Using the we have:
∫(1 dx) ≈ (Δx/2) * [f(x0) + 2 * (f(x1) + f(x2) + ... + f(xn-1)) + f(xn)]
where Δx = (b - a) / n is the interval width, and f(x) = 1.
In this case, a = 2, b = 10, and n = 10.
Δx = (10 - 2) / 10 = 8 / 10 = 0.8
x0 = 2
x1 = 2 + 0.8 = 2.8x2 = 2.8 + 0.8 = 3.6
...xn = 10
Plugging these values into the trapezoidal rule formula:
∫(1 dx) ≈ (0.8/2) * [1 + 2 * (1 + 1 + ... + 1) + 1] ≈ (0.8/2) * [1 + 2 * 9 + 1] ≈ (0.8/2) * 19 ≈ 7.6
So, using the trapezoidal rule, the approximate value of the integral is 7.6.
b) Midpoint Rule:
The midpoint rule approximates the integral by evaluating the function at the midpoint of each interval and multiplying it by the width of the interval.
Using the midpoint rule, we have:
∫(1 dx) ≈ Δx * [f((x0 + x1)/2) + f((x1 + x2)/2) + ... + f((xn-1 + xn)/2)]
In this case, using the same values for a, b, and n as before, we have:
Δx = 0.8
Using the midpoint rule formula:
∫(1 dx) ≈ 0.8 * [1 + 1 + ... + 1] ≈ 0.8 * 10 ≈ 8
So, using the midpoint rule, the approximate value of the integral is 8.
c) Simpson's Rule:Simpson's rule approximates the integral using quadratic polynomials.
Using Simpson's rule, we have:
∫(1 dx) ≈ (Δx/3) * [f(x0) + 4 * f(x1) + 2 * f(x2) + 4 * f(x3) + ... + 2 * f(xn-2) + 4 * f(xn-1) + f(xn)]
In this case, using the same values for a, b, and n as before, we have:
Δx = 0.8
Using Simpson's rule formula:
∫(1 dx) ≈ (0.8/3) * [1 + 4 * 1 + 2 * 1 + 4 * 1 + ... + 2 * 1 + 4 * 1 + 1] ≈ (0.8/3) * [1 + 4 * 9 + 1] ≈ (0.8/3) * 38 ≈ 10.133333333
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Use the Ratio Test to determine whether the series is convergent or divergent. n gn n=1 Identify an Evaluate the following limit. an + 1 lim an n-00 Since lim n- an + 1 an 1, the series is convergent
By applying the Ratio Test to the series, we can determine its convergence or divergence. Given that the limit of (an+1 / an) as n approaches infinity is less than 1, the series is convergent.
The Ratio Test is a method used to determine the convergence or divergence of a series. For a series ∑gn, where gn is a sequence of terms, the Ratio Test involves evaluating the limit of the ratio of consecutive terms, (gn+1 / gn), as n approaches infinity.
In this case, we have a series with terms represented as an. To apply the Ratio Test, we evaluate the limit of (an+1 / an) as n approaches infinity. Given that the limit is less than 1, specifically equal to 1, it indicates convergence. This can be seen from the statement that lim n→∞ (an+1 / an) = 1.
When the limit of the ratio is less than 1, it implies that the series converges absolutely. The series becomes smaller and smaller as n increases, indicating that the sum of the terms approaches a finite value. Therefore, based on the result of the Ratio Test, we can conclude that the series is convergent.
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Let F = (9x²y + 3y3 + 2er)i + (3ev? + 225x) ;. Consider the line integral of F around the circle of radius a, centered at the origin and traversed counterclockwise. (a) Find the line integral for a = 1. line integral = (b) For which value of a is the line integral a maximum?
The value of a that maximizes the line integral is 15√3/2. Line integrals are a concept in vector calculus that involve calculating the integral of a vector field along a curve or path.
To evaluate the line integral of the vector field F around the circle of radius a centered at the origin and traversed counterclockwise, we can use Green's theorem. Green's theorem states that the line integral of a vector field around a closed curve is equal to the double integral of the curl of the vector field over the region enclosed by the curve.
Given vector field F = (9x²y + 3y³ + 2er)i + (3ev? + 225x)j, we can calculate its curl:
curl(F) = ∇ x F
= (∂/∂x, ∂/∂y, ∂/∂z) x (9x²y + 3y³ + 2er, 3ev? + 225x)
= (0, 0, (∂/∂x)(3ev? + 225x) - (∂/∂y)(9x²y + 3y³ + 2er))
= (0, 0, 225 - 6y² - 6y)
Since the curl has only a z-component, we can ignore the first two components for our calculation.
Now, let's evaluate the double integral of the z-component of the curl over the region enclosed by the circle of radius a centered at the origin.
∬ R (225 - 6y² - 6y) dA
To find the maximum value of the line integral, we need to determine the value of a that maximizes this double integral. Since the region enclosed by the circle is symmetric about the x-axis, we can integrate over only the upper half of the circle.
Using polar coordinates, we have:
x = rcosθ
y = rsinθ
dA = r dr dθ
The limits of integration for r are from 0 to a, and for θ from 0 to π.
∫[0,π]∫[0,a] (225 - 6r²sin²θ - 6r sinθ) r dr dθ
Let's solve this integral to find the line integral for a = 1.
The integral can be split into two parts:
∫[0,π]∫[0,a] (225r - 6r³sin²θ - 6r² sinθ) dr dθ
= ∫[0,π] [(225/2)a² - (6/4)a⁴sin²θ - (6/3)a³sinθ] dθ
= π[(225/2)a² - (6/4)a⁴] - 6π/3 [(a³/3 - a³/3)]
= π[(225/2)a² - (6/4)a⁴ - 6/3a³]
Substituting a = 1, we get:
line integral = π[(225/2) - (6/4) - 6/3]
= π[112.5 - 1.5 - 2]
= π(109)
Therefore, the line integral for a = 1 is 109π.
To find the value of a that maximizes the line integral, we can take the derivative of the line integral with respect to a and set it equal to zero.
d(line integral)/da = 0
Differentiating π[(225/2)a² - (6/4)a⁴ - 6/3a³] with respect to a, we have:
π[225a - (6/2)4a³ - (6/3)3a²] = 0
225a - 12a³ - 6a² = 0
a(225 - 12a² - 6a) = 0
The values of a that satisfy this equation are a = 0, a = ±√(225/12).
However, a cannot be negative or zero since it represents the radius of the circle, so we consider only the positive value:
a = √(225/12) = √(225)/√(12) = 15/√12 = 15√3/2
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hint For normally distributed data, what proportion of observations have a z-score greater than 1.92. Round to 4 decimal places.
Approximately 0.0274, or 2.74%, of observations have a z-score greater than 1.92.
In a normal distribution, the z-score represents the number of standard deviations a particular observation is away from the mean. To find the proportion of observations with a z-score greater than 1.92, we need to calculate the area under the standard normal curve to the right of 1.92.
Using a standard normal distribution table or a statistical software, we can find that the area to the right of 1.92 is approximately 0.0274. This means that approximately 2.74% of observations have a z-score greater than 1.92.
This calculation is based on the assumption that the data follows a normal distribution. The proportion may vary if the data distribution deviates significantly from normality. Additionally, it's important to note that the specific proportion will depend on the level of precision required, as rounding to four decimal places introduces a small level of approximation
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1. If f(x) = 5x¹ - 6x² + 4x - 2, find f'(x) and f'(2). STATE all rules used.
Rules used in the above solution are: Power Rule, Sum Rule, Constant Rule, and Subtraction Rule.
Given function: f(x) = 5x¹ - 6x² + 4x - 2We are supposed to find f'(x) and f'(2).f'(x) is the derivative of the function f(x). The derivative of any polynomial is found by differentiating each of its terms.
Now, let us find f'(x):f'(x) = d/dx (5x¹) - d/dx (6x²) + d/dx (4x) - d/dx (2)f'(x) = 5 - 12x + 4f'(x) = 9 - 12x
Now, we have f'(x) = 9 - 12x.
We have to find f'(2) which means we substitute x = 2 in f'(x):f'(2) = 9 - 12(2)f'(2) = 9 - 24f'(2) = -15
Therefore, the derivative of the given function is 9 - 12x and the value of f'(2) is -15. Rules used in the above solution are: Power Rule, Sum Rule, Constant Rule, and Subtraction Rule.
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Binomial -- A certain type of fuel pump has been installed on n airliners. An airliner has only one
fuel pump. The pump has a defect that might cause it to fail in flight. I = probability a pump fails.
1) Suppose the probability of failure is n = 0.13 and the pump is installed on n = 11 airliners.
What is the probability that 3 airliners suffer a pump failure?
• Prob. = 0.119
2) If probability of failure is n = 0.30 and the pump is installed on n = 11 airliners, what is the
probability that 5 or more airliners suffer a pump failure?
Prob. = 0.210 3) If the probability of failure is m = 0.25 and the pump is installed on n = 36 airliners, what is the
probability that 12 or fewer airliners suffer a pump failure?
The probability that 5 or more airliners suffer a pump failure is approximately 0.210.
1) using the binomial distribution with n = 11 (number of airliners) and p = 0.13 (probability of failure), we can calculate the probability that exactly 3 airliners suffer a pump failure. the formula for this probability is p(x = k) = c(n, k) * pᵏ * (1 - p)⁽ⁿ ⁻ ᵏ⁾, where c(n, k) is the binomial coefficient.using this formula, we find:p(x = 3) = c(11, 3) * 0.13³ * (1 - 0.13)⁽¹¹ ⁻ ³⁾
= 165 * 0.13³ * 0.87⁸ ≈ 0.119therefre, the probability that exactly 3 airliners suffer a pump failure is approximately 0.119.
2) to find the probability that 5 or more airliners suffer a pump failure, we need to calculate the cumulative probability p(x ≥ 5). we can do this by finding the probabilities of 5, 6, 7, ..., 11 failures and summing them up.using the binomial distribution with n = 11 and p = 0.30, we find:
p(x ≥ 5) = p(x = 5) + p(x = 6) + ... + p(x = 11) ≈ 0.210
3) using the binomial distribution with n = 36 (number of airliners) and p = 0.25 (probability of failure), we can calculate the probability that 12 or fewer airliners suffer a pump failure. to find this probability, we need to sum the probabilities of 0, 1, 2, ..., 12 failures.using the binomial distribution formula, we find:
p(x ≤ 12) = p(x = 0) + p(x = 1) + ... + p(x = 12)calculating this sum will give us the probability that 12 or fewer airliners suffer a pump failure.
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A land parcel has topographic contour of an area can be mathematically
represented by the following equation:
2 = 0.5x4 + xIny + 2cox For earthwork purpose, the landowner needs to know the contour
slope with respect to each independent variables of the contour.
Determine the slope equations.
Compute the contour slopes in x and y at the point (2, 3).
The contour slope in x at point (2,3) is given by 16.6337+2c cos(2), and the contour slope in y at point (2,3) is given by 0.2397.
In order to find the slope equations for a land parcel with topographic contour, we first need to identify the independent variables involved in the contour equation given.
In this case, the independent variables are x and y.
The slope equation for the variable x can be found by taking the partial derivative of the contour equation with respect to x.
This is given as follows: [tex]$$\frac{\partial z}{\partial x}=2x^3+\frac{y}{x\ln(10)}+2c\cos(x)=f_x(x,y)$$[/tex]
Similarly, the slope equation for the variable y can be found by taking the partial derivative of the contour equation with respect to y.
This is given as follows: [tex]$$\frac{\partial z}{\partial y}=\frac{x}{y\ln(10)}=f_y(x,y)$$[/tex]
Now that we have the slope equations, we can compute the contour slopes in x and y at the point (2,3) as follows:
At point (2,3), x = 2 and y = 3.
Therefore, the slope equation for x becomes: [tex]$$f_x(2,3)=2(2)^3+\frac{3{2\ln(10)}+2c\cos(2)=16.6337+2c\cos(2)$$[/tex]
Similarly, the slope equation for y becomes: [tex]$$f_y(2,3)=\frac{2}{3\ln(10)}=0.2397$$[/tex]
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show steps!
find the radius and the interval of convergence of the
series:
E (summation/sigma symbol) (with infinity sign on top and on the
bottom of the symbol, it is k=1) (x-2)^k / k*4^k.
The radius of convergence of the series is 4 and the interval of convergence is (-2, 6).
To find the radius of convergence, we can use the ratio test. According to the ratio test, if we take the limit as k approaches infinity of the absolute value of the ratio of the (k+1)th term to the kth term, and this limit is less than 1, then the series converges.
Let's apply the ratio test to the given series:
lim(k→∞) |((x-2)^(k+1))/(k+1)*(4^(k+1))| / |((x-2)^k)/(k*4^k)|
Simplifying this expression, we get:
lim(k→∞) |(x-2)/(k+1)| * |4/4|
Taking the absolute value and simplifying further, we have:
lim(k→∞) |x-2|/|k+1|
To ensure that this limit is less than 1, we need |x-2| < |k+1|.
Since |k+1| increases as k increases, we need |x-2| < |k+1| to hold true for all values of k.
Therefore, the radius of convergence is determined by the inequality |x-2| < |k+1|, which means the series converges for values of x that are within a distance of 4 units from the center x = 2. Thus, the radius of convergence is 4.
The interval of convergence can be found by considering the values of x that satisfy the inequality |x-2| < 4. Solving this inequality, we have -2 < x-2 < 2, which gives -2 < x < 4. Therefore, the interval of convergence is (-2, 4).
In summary, the series has a radius of convergence of 4 and an interval of convergence of (-2, 4).
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1. Find the arc length of the cardioid: r=1+ cos 0 2. Find the area of the region inside r = 1 and inside the region r = 1 + cos2 3. Find the area of the four-leaf rose: r = 2 cos(20)
trigonometric identities, we know that cos²(θ) = (1 + cos(2θ))/2. Applying this identity:
A = (1/2)∫[0,2π] 4(1 + cos(40))/2 dθ
A = 2π(1 + cos(40))
Evaluating the integral will give us the area of the four-leaf rose.
1. To find the arc length of the cardioid given by the equation r = 1 + cos(θ), we can use the arc length formula in polar coordinates:
L = ∫√(r² + (dr/dθ)²) dθ
Here, r = 1 + cos(θ), so we need to find dr/dθ:
dr/dθ = -sin(θ)
Substituting these values into the arc length formula, we have:
L = ∫√((1 + cos(θ))² + (-sin(θ))²) dθ = ∫√(1 + 2cos(θ) + cos²(θ) + sin²(θ)) dθ
= ∫√(2 + 2cos(θ)) dθ
This integral can be evaluated using appropriate techniques such as substitution or trigonometric identities.
provide the arc length of the cardioid.
2. To find the area of the region inside r = 1 and inside the region r = 1 + cos²(θ), we can set up the double integral:
A = ∬D r dr dθ
where D represents the region of interest .
In this case, the region D is defined by the conditions 0 ≤ r ≤ 1 + cos²(θ) and 0 ≤ θ ≤ 2π.
To evaluate the integral, we can convert to Cartesian coordinates using the transformation equations x = rcos(θ) and y = rsin(θ). The limits of integration for x and y will then depend on the polar coordinates.
The integral expression will be:
A = ∫∫D dA = ∫∫D dx dy
where D is the region defined by the given conditions. Evaluating this integral will give us the area of the region.
3. The area of the four-leaf rose given by the equation r = 2cos(2θ) can be found using the formula for the area in polar coordinates:
A = (1/2)∫[a,b] (r²) dθ
In this case, r = 2cos(20), so we substitute this into the formula:
A = (1/2)∫[0,2π] (2cos(20))² dθ
Simplifying further:
A = (1/2)∫[0,2π] 4cos²(20) dθ
Using
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find the centroid of the region bounded by the given curves. y = 2 sin(3x), y = 2 cos(3x), x = 0, x = 12 (x, y) =
The volume of the solid obtained by rotating the region bounded by the curves y = 4 sec(x), y = 6, and −3 ≤ x ≤ 3 about the line y = 4 is approximately X cubic units.
To find the volume, we can use the method of cylindrical shells. The region bounded by the curves y = 4 sec(x), y = 6, and −3 ≤ x ≤ 3 is a region in the xy-plane. When this region is rotated about the line y = 4, it creates a solid with a cylindrical shape. We can imagine dividing this solid into thin vertical slices or cylindrical shells.
The height of each cylindrical shell is given by the difference between the y-coordinate of the curve y = 6 and the y-coordinate of the curve y = 4 sec(x), which is 6 - 4 sec(x). The radius of each cylindrical shell is the distance between the line y = 4 and the curve y = 4 sec(x), which is 4 sec(x) - 4.
To calculate the volume of each cylindrical shell, we multiply its height by its circumference (2π times the radius). Integrating the volume of all these cylindrical shells over the range of x from −3 to 3 gives us the total volume of the solid.
Performing the integration and evaluating it will give us the numerical value of the volume, which is X cubic units.
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Find the value of f(5) (1) if f(x) is approximated near x = 1 by the Taylor polynomial 10 p(x) = [ (x −1)n n=0 n!
The value of f(5) using Taylor Polynomial is 0.0007031250.
1. Determine the degree of the Taylor Polynomial p(x).
In this case, the degree of the Taylor polynomial is 10, since p(x) is equal to (x-1)10.
2. Calculate the value of f(5) using the formula for the Taylor polynomial.
f(5) = 10 ∑ [(5 - 1)n/ n!]
= 10 ∑ [(4/ n!
= 10[(4 + (4)2/2! + (4)3/3! + (4)4/4! + (4)5/5! + (4)6/6! + (4)7/7! + (4)8/8! + (4)9/9! + (4)10/10!]
= 10[256/3628800]
= 0.0007031250
Therefore, the value of f(5) is 0.0007031250.
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The integral 7√1 - 4x² dx is to be evaluated directly and using a series approximation. (Give all your answers rounded to 3 significant figures.) a) Evaluate the integral exactly, using a substitut
To evaluate the integral ∫(7√(1 - 4x²)) dx exactly, a substitution method can be used. The substitution u = 1 - 4x² is made, which simplifies the integral to ∫(7√u) dx. The integral is then evaluated in terms of u and x.
To evaluate the integral ∫(7√(1 - 4x²)) dx exactly, we can make a substitution u = 1 - 4x². Taking the derivative of u with respect to x, du/dx = -8x. Solving for dx, we get dx = du / (-8x).
Now, substituting these values into the original integral, we have ∫(7√u) (du / (-8x)). Since u = 1 - 4x², we can express x in terms of u as x = ±√((1 - u) / 4). Substituting this into the integral, we obtain ∫((7√u) (du / (-8(±√((1 - u) / 4)))).
Simplifying further, the integral becomes ∫(-7√u / (8√(1 - u))) du. To solve this integral, we can use the substitution v = 1 - u. Differentiating v with respect to u, dv/du = -1. Rearranging, we get du = -dv. Substituting these values into the integral, we have ∫(-7√v / (8√v)) (-dv) = ∫(7√v / (8√v)) dv.
Integrating √v / √v, we get ∫(7/8) dv = (7/8)v + C, where C is the constant of integration. Replacing v with 1 - u, we finally obtain the exact integral as (7/8)(1 - u) + C.
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Evaluate the integral using integration by parts. Do not use any other method. You must show your work. Vu x sin(x) dx
Integration by parts method is a method of integration that involves choosing one part of the function as the “first” function and the remaining part of the function as the “second” function.
The integral of the product of these functions can be calculated using the integration by parts formula.
Let us evaluate the integral:
∫v(x)sin(x)dx
Let us assume that
u(x) = sin(x), then,
dv(x)/dx = v(x) = v = x
To integrate the above integral using the integration by parts formula:
∫u(x)dv(x) = u(x)v - ∫v(x)du(x)/dx dx
Thus, substituting the value of u(x) and dv(x), we get:
∫sin(x)x dx = sin(x) ∫x dx - ∫ (dx/dx) (x cos(x)) dx
= -x cos(x) + sin(x) + C,
where C is the constant of integration.
Therefore, the integral using integration by parts is given by-
∫x cos(x) dx = x sin(x) - ∫sin(x) dx= -x cos(x) + sin(x) + C,
where C is the constant of integration.
Final Answer: Therefore, the integral using integration by parts is given by- ∫x cos(x) dx = -x cos(x) + sin(x) + C.
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Use the product to sum formula to fill in the blanks in the identity below: sin(82)cos(2x) - ( 1 (sin( 2 2) + sin( 2) Put the smaller number in the first box. Use half angle formulas or formula for"
Using the product-to-sum formula, the identity can be filled in as follows: sin(82)cos(2x) - (1/2)(sin(4) + sin(2)).
The product-to-sum formula states that sin(A)cos(B) = (1/2)[sin(A + B) + sin(A - B)]. In the given identity, we have sin(82)cos(2x). By comparing it with the formula, we can see that A = 82 and B = 2x. Applying the formula, we get (1/2)[sin(82 + 2x) + sin(82 - 2x)].
The next part of the identity is -(1/2)(sin(22) + sin(2)). To match this with the product-to-sum formula, we need to rewrite the angles in terms of the sum and difference. We have 22 = 4 + 18 and 2 = 4 - 2. Plugging these values into the formula, we get -(1/2)[sin(4 + 18) + sin(4 - 2)], which simplifies to -(1/2)(sin(22) + sin(2)).
Combining both parts, the identity becomes sin(82)cos(2x) - (1/2)[sin(82 + 2x) + sin(82 - 2x)] - (1/2)(sin(22) + sin(2)).
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uppose that the number of bacteria in a certain population increases according to a continuous exponential growth model. A sample of 3000 bacteria selected from this population reached the size of 3622 bacteria in six hours. Find the hourly growth rate parameter.
the hourly growth rate parameter is approximately 0.0381, indicating that the population of bacteria is increasing by approximately 0.0381 per hour according to the continuous exponential growth model.
In this case, the initial population size A₀ is 3000 bacteria, the final population size A is 3622 bacteria, and the time period t is 6 hours. We want to find the growth rate parameter k.
Using the formula A = A₀ × [tex]e^(kt)[/tex], we can rearrange the equation to solve for k:
k = (1/t) × ln(A/A₀)
Substituting the given values:
k = (1/6) × ln(3622/3000) ≈ 0.0381 per hour
Therefore, the hourly growth rate parameter is approximately 0.0381, indicating that the population of bacteria is increasing by approximately 0.0381 per hour according to the continuous exponential growth model.
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A ladder 10 ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a speed of 2 ft/sec, how fast is the angle between the top of the ladder and the wall changing when the angle is radians?
When the angle between the top of the ladder and the wall is θ = π/4 radians, the angle is changing at a rate of -2√2 ft/sec.
Let's denote the length of the ladder as L (10 ft) and the distance from the bottom of the ladder to the wall as x. The height of the ladder from the ground is h, and the angle between the ladder and the wall is θ. We can use the Pythagorean theorem to relate the variables:
x^2 + h^2 = L^2
Differentiating both sides of the equation with respect to time t, we get:
2x(dx/dt) + 2h(dh/dt) = 0
Since the bottom of the ladder slides away from the wall at a speed of 2 ft/sec, we have dx/dt = 2 ft/sec.
We are interested in finding how fast the angle θ is changing, so we need to determine dh/dt when θ = π/4 radians.
At θ = π/4 radians, we have:
x = h (since it is an isosceles right triangle)
x^2 + x^2 = L^2
2x^2 = L^2
x = L/√2
Substituting this value of x into the differentiated equation, we have:
2(L/√2)(dx/dt) + 2h(dh/dt) = 0
(L)(2)(2) + 2h(dh/dt) = 0
4L + 2h(dh/dt) = 0
Solving for dh/dt, we get:
2h(dh/dt) = -4L
dh/dt = -2L/h
At θ = π/4 radians, h = x = L/√2, so:
dh/dt = -2L/(L/√2)
dh/dt = -2√2 ft/sec
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Someone can help me to solve this problem? show all steps
please!
= - Problem 10. Consider the vector valued function F(x, y, z) = (y sin(x2 + y²), -x sin(x2 + y2), z(3 – 2y)) and the region W = {(x,y,z) € R3 : 22 + y2 + x2 0}. Compute Saw F. = :
After considering the given data we conclude that the value of the[tex]\int _{aw} F[/tex] is [tex](4/15) \pi[/tex], under the condition that [tex]W = {(x,y,z) \in R^3 : x^2 + y^2 + z^2\leq 1, z \geq 0}.[/tex] using the divergence theorem.
To find the value of the integral [tex]\int _{aw} F[/tex], we need to apply the divergence theorem, which relates the surface integral of the normal component of a vector field over a closed surface to the volume integral of the divergence of the vector field over the region enclosed.
Let's first compute the divergence of F:
[tex]F = (\sigma/\sigma x)(y sin(x^2 + y^2)) + (\sigma/\sigma y)(-x sin(x^2 + y^2)) + (\sigma/\sigma z)(z(3 - 2y))= 2xy cos(x^2 + y^2) - z(2)[/tex]
Next, we need to find a closed surface that encloses the region W. Since W is a hemisphere of radius 1 centered at the origin, we can use the upper hemisphere of radius 1 as our closed surface. Let S be the surface of the hemisphere, oriented outward. Then, by the divergence theorem, we have:
[tex]\int _{aw} F = \int ^S _F * n dS = \int _S (F1, F2, F3) *(0, 0, 1) dS[/tex]
where n is the unit normal vector to the surface S, pointing outward.
Since the surface S is a hemisphere of radius 1 centered at the origin, we can parameterize it as:
[tex]x = sin \theta cos \varphi[/tex]
[tex]y = sin \theta sin \varphi[/tex]
[tex]z = cos \theta[/tex]
[tex]where 0 \leq \theta \leq \pi/2 and 0 \leq \varphi \leq 2\pi.[/tex]
Then, the unit normal vector to the surface S is given by:
[tex]n = (sin \theta cos \varphi, sin \theta sin \varphi, cos \theta)[/tex]
Therefore, we have:
[tex]F * n = (y sin(x^2 + y^2), -x sin(x^2 + y^2), z(3 - 2y)) *(sin \theta cos \varphi, sin \theta sin \varphi, cos \theta)[/tex]
[tex]= y sin(x^2 + y^2) sin \theta cos \varphi - x sin(x^2 + y^2) sin \theta sin \varphi + z(3 - 2y) cos \theta[/tex]
[tex]= sin \theta cos \varphi sin(\theta^2 cos \varphi^2 + \theta^2 sin \varphi^2) - sin \theta sin \varphi sin(\theta^2 cos \varphi^2 + \theta^2 sin \varphi^2) + cos \theta (3 - 2y)z[/tex]
[tex]= cos \theta (3 - 2y)z[/tex]
Therefore, we have:
[tex]\int _{aw} F = \int ^S_ F * n dS = \int _0^2\pi \int _0^ {\pi/2} cos \theta (3 - 2y)z sin \theta d\theta d\varphi[/tex]
To evaluate this integral, we can use the substitution [tex]x = sin \theta, dx = cos \theta d\theta,[/tex] and the fact that the volume of the hemisphere of radius 1 is [tex](2/3)\pi[/tex]. Then, we get:
[tex]\int _{aw} F = \int _0^{2\pi} \int _0^1 (3 - 2y)z x^2 dx d\varphi[/tex]
[tex]= (2/3)\pi \int _0^1 (3 - 2y)z y^2 dy[/tex]
To evaluate this integral, we need to know the function z(y) that describes the upper half of the sphere of radius 1. Since z ≥ 0, we have z [tex]= \sqrt(1 - x^2 - y^2), so z = \sqrt(1 - y^2)[/tex] for the upper half of the sphere. Therefore, we get:
[tex]\int _{aw} F = (2/3)\pi \int _0^1 (3 - 2y) \sqrt(1 - y^2) y^2 dy[/tex]
This integral can be evaluated using the substitution[tex]u = 1 - y^2, du = -2y dy,[/tex] and the fact that the integral of[tex]u^{(3/2) }[/tex]is [tex](2/5)u^{(5/2)}.[/tex] After some algebraic manipulation, we get:
[tex]\int _{aw} F = (4/15)\pi[/tex]
Therefore, the value of the integral [tex]\int _{aw} F is (4/15)\pi.[/tex]
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The complete question is
Consider the vector valued function F(x, y, z) = (y sin(x2 + y²), -x sin(x2 + y2), z(3 – 2y)) and the region W = {(x,y,z) € R³ : x² + y² + z²≤ 1, z ≥0}. Compute \int _aw F. = :
meredith is a general surgeon who performs surgeries such as appendectomies and laparoscopic cholecystectomies. the average number of sutures that meredith uses to close a patient is 37, and the standard deviation is 8. the distribution of number of sutures is right skewed. random samples of 32 are drawn from meredith's patient population, and the number of sutures used to close each patient is noted. use the central limit theorem to find the mean and standard error of the sampling distribution. select the statement that describes the shape of the sampling distribution. group of answer choices unknown the sampling distribution is normally distributed with a mean of 37 and standard deviation 1.41. the sampling distribution is right skewed with a mean of 37 and standard deviation 8. the sampling distribution is normally distributed with a mean of 37 and standard deviation 8. the sampling distribution is right skewed with a mean of 37 and standard deviation 1.41.
The statement that accurately describes the form of the sampling distribution is:The inspecting dissemination is regularly circulated with a mean of 37 and standard deviation 1.41.
According to the central limit theorem, regardless of how the population distribution is shaped, the sampling distribution of the sample mean will be approximately normally distributed for a sufficiently large sample size.
For this situation, irregular examples of 32 are drawn from Meredith's patient populace, which fulfills the state of a sufficiently huge example size. The central limit theorem can be used to determine the sampling distribution's mean and standard error.
In this instance, the population mean, which is 37, is equal to the mean of the sampling distribution.
The population standard deviation divided by the square root of the sample size is the sampling distribution's standard error. For this situation, the standard mistake is 8 partitioned by the square foundation of 32, which is around 1.41.
Therefore, the statement that accurately describes the form of the sampling distribution is:
The inspecting dissemination is regularly circulated with a mean of 37 and standard deviation 1.41.
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Urgent please help!!
Upon the death of his uncle, Lucien receives an inheritance of $50,000, which he invests for 15 years at 6.9%, compounded continuously. What is the future value of the inheritance? The future value is
The future value of the inheritance is approximately $137,396.32.
To find the future value of the inheritance, we can use the continuous compound interest formula:
P = Po * e^(kt)
Where:
P = Future value
Po = Present value (initial investment)
k = Interest rate (in decimal form)
t = Time period (in years)
e = Euler's number (approximately 2.71828)
Po = $50,000
k = 6.9% = 0.069 (in decimal form)
t = 15 years
Plugging in these values into the formula, we get:
P = 50000 * e^(0.069 * 15)
Calculating this using a calculator or computer software, the future value of the inheritance is approximately $137,396.32.
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(1 point) Find the directional derivative of f(x, y, z)=z³ - x²y at the point (-3, 1, -2) in the direction of the vector v = (5, 1, -1).
To find the directional derivative of the function f(x, y, z) = z³ - x²y at the point (-3, 1, -2) in the direction of the vector v = (5, 1, -1), we can use the gradient operator.
The gradient of a function f(x, y, z) is defined as:
∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z)
First, let's calculate the partial derivatives of f(x, y, z):
∂f/∂x = -2xy
∂f/∂y = -x²
∂f/∂z = 3z²
Now, evaluate these partial derivatives at the point (-3, 1, -2):
∂f/∂x = -2(-3)(1) = 6
∂f/∂y = -(-3)² = -9
∂f/∂z = 3(-2)² = 12
The gradient of f(x, y, z) at the point (-3, 1, -2) is therefore:
∇f = (6, -9, 12)
To find the directional derivative, we take the dot product of the gradient and the unit vector in the direction of v.
First, we need to normalize the vector v to obtain the unit vector u:
||v|| = √(5² + 1² + (-1)²) = √27 = 3√3
The unit vector u in the direction of v is:
u = v / ||v|| = (5/3√3, 1/3√3, -1/3√3)
Now, we can calculate the directional derivative:
D_v f = ∇f · u = (6, -9, 12) · (5/3√3, 1/3√3, -1/3√3)
D_v f = (6 * 5/3√3) + (-9 * 1/3√3) + (12 * -1/3√3)
= 10/√3 - 3/√3 - 4/√3
= (10 - 3 - 4)/√3
= 3/√3
= √3
Therefore, the directional derivative of f(x, y, z) = z³ - x²y at the point (-3, 1, -2) in the direction of the vector v = (5, 1, -1) is √3.
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3. To find the surface area of the part of the paraboloid
z=9−x2−y2 above the plane z=5 , what would be the projection region
(region of integration) on the xy-plane?
4. Finding the surface area Question 3 1 pts = To find the surface area of the part of the paraboloid z = 9 – x2 - y2 above the plane z= 5, what would be the projection region (region of integration) on the xy-plane? A disk of
The projection region on the xy-plane for the part of the paraboloid [tex]z = 9 - x^2 - y^2[/tex] above the plane z = 5 is a disk.
To understand why the projection region is a disk, we need to consider the equations of the surfaces involved. The equation z = 5 represents a horizontal plane parallel to the xy-plane, located at a height of 5 units above the origin.
The equation of the paraboloid, [tex]z = 9 - x^2 - y^2[/tex], represents an upward-opening parabolic surface centered at the origin. The region of interest is the part of the paraboloid that lies above the plane z = 5.
To determine the projection region on the xy-plane, we set z = 5 in the equation of the paraboloid:
[tex]5 = 9 - x^2 - y^2[/tex]
Rearranging the equation, we have:
[tex]x^2 + y^2 = 4[/tex]
This equation represents a circle centered at the origin with a radius of 2 units. Therefore, the projection region on the xy-plane is a disk of radius 2 units.
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Find the following derivative using the Product or Quotient Rule: 2 d X² dx 3x + 7 In your answer: • Describe what rules you need to use, and give a short explanation of how you knew that the rule was relevant here. Label any intermediary pieces or parts. Show some work to demonstrate that you know how to apply the derivative rules you're talking about. • State your answer
The derivative of the function d(x² + 3x + 7)/dx is 2x + 3
How to find the derivative of the functionFrom the question, we have the following parameters that can be used in our computation:
The function x² + 3x + 7
This can be expressed as
d(x² + 3x + 7)/dx
The derivative of the function can be calculated using the first principle which states that
if f(x) = axⁿ, then f'(x) = naxⁿ⁻¹
Using the above as a guide, we have the following:
d (x² + 3x + 7)/dx = 2x + 3
Hence, the derivative is 2x + 3
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Question
Find the following derivative using the Product or Quotient Rule:
d(x² + 3x + 7)/dx
In your answer: • Describe what rules you need to use, and give a short explanation of how you knew that the rule was relevant here. Label any intermediary pieces or parts. Show some work to demonstrate that you know how to apply the derivative rules you're talking about. • State your answer
(1 point) Find the following integral. Note that you can check your answer by differentiation. 6e2vý dy = VÝ
The integral of 6e^(2vy) dy is 3e^(2vy) + C, where C is the constant of integration. This answer can be verified by differentiating 3e^(2vy) + C with respect to y,
The given integral is 6e^(2vy) dy. To integrate this expression, use the formula:integral e^(ax)dx=1/a * e^(ax)where a is a constant and dx is the differential of x.According to this formula, we can rewrite the given integral as:∫ 6e^(2vy) dy = 6 * 1/2 * e^(2vy) + C = 3e^(2vy) + Cwhere C is the constant of integration.To check this answer by differentiation, differentiate the expression 3e^(2vy) + C with respect to y, we get:d/dy [3e^(2vy) + C] = 3 * 2v * e^(2vy) + 0 = 6ve^(2vy)which is equal to the integrand 6e^(2vy). Therefore, our answer is correct.
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Which of these four sets of side lengths will form a right triangle?
Set 1,
√√2 cm, 9 cm, 7 cm
Set 3
6 mm, 2 mm, 10 mm
Set 2
2 in., √√5 in., 9 in.
Set 4
√√2 tt. √√7 ft. 3 ft
Set 3 (6 mm, 2 mm, 10 mm) is the only set of side lengths that forms a right triangle.
We have,
To determine whether a set of side lengths will form a right triangle, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.
Let's examine each set of side lengths:
Set 1:
√√2 cm, 9 cm, 7 cm
To determine if it forms a right triangle, we need to check if the Pythagorean theorem holds:
(√√2)² + 7² = 9²
2 + 49 ≠ 81
Therefore, Set 1 does not form a right triangle.
Set 3:
6 mm, 2 mm, 10 mm
Applying the Pythagorean theorem:
6^2 + 2^2 = 10^2
36 + 4 = 100
Therefore, Set 3 forms a right triangle.
Set 2:
2 in, √√5 in., 9 in.
Using the Pythagorean theorem:
2² + (√√5)² ≠ 9²
Hence, Set 2 does not form a right triangle.
Set 4:
√√2 tt., √√7 ft., 3 ft
To apply the Pythagorean theorem, we need to convert the side lengths to a consistent unit:
√√2 tt. = √√2 x 12 in.
√√7 ft. = √√7 x 12 in.
3 ft. = 3 x 12 in.
Then, we can check:
(√√2 x 12)² + (√√7 x 12)² ≠ (3 x 12)²
Therefore, Set 4 does not form a right triangle.
Thus,
Set 3 (6 mm, 2 mm, 10 mm) is the only set of side lengths that forms a right triangle.
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Find the relative maximum and minimum values. 2 2 f(x,y) = x² + y² = x² + y² - 6x +10y - 9 Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. The function has a relative maximum value of f(x,y) = at (x,y) = (Simplify your answers. Type exact answers. Type an ordered pair in the second answer box.) B. The function has no relative maximum value.
The function f(x, y) = x² + y² - 6x + 10y - 9 does not have a relative maximum value.
To determine the relative maximum and minimum values of a function, we need to analyze its critical points and evaluate the function at those points. Critical points occur where the partial derivatives with respect to x and y are equal to zero or do not exist.
Taking the partial derivative of f(x, y) with respect to x, we have:
∂f/∂x = 2x - 6
Taking the partial derivative of f(x, y) with respect to y, we have:
∂f/∂y = 2y + 10
To find the critical points, we set these partial derivatives equal to zero and solve the resulting equations:
2x - 6 = 0 => x = 3
2y + 10 = 0 => y = -5
Therefore, the only critical point is (3, -5).
To determine if this critical point is a relative maximum or minimum, we can use the second partial derivative test or evaluate the function at surrounding points. However, since the function has no terms involving xy, the second partial derivative test is inconclusive.
We can examine the values of f(x, y) at the critical point and some nearby points. Evaluating f(x, y) at (3, -5), we get:
f(3, -5) = (3)² + (-5)² - 6(3) + 10(-5) - 9 = 0
Since the value of f(x, y) at the critical point is 0, we conclude that there is no relative maximum value for the function. Therefore, the correct choice is B: The function has no relative maximum value.
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Use geometry (not Riemann sums) to evaluate the following definite integral. Sketch a graph of the integrand, show the region in question, and interpret your results. 4 5 if x < 3 Inoncen f(x)dx, wher
Given an integral∫_4^5▒〖f(x)dx 〗 where f(x) is defined as follows:
For x < 3, f(x) = 0
For x ≥ 3, f(x) = x - 3
The graph of the integrand is shown below:
This is a piecewise function defined on the interval [4, 5].
It is zero for x < 3, and for x ≥ 3 it is equal to x - 3.
We can graph the two parts of the function separately, and then find their areas, which will give us the value of the integral.
To graph the function, we first draw a vertical line at x = 3, which separates the function into two parts.
For x < 3, we draw a horizontal line at y = 0, which is the x-axis.
For x ≥ 3, we draw a line with a slope of 1, which passes through the point (3, 0).
This line has the equation y = x - 3, and it is shown in blue in the graph above.
The region in question is the shaded region between the graph of the integrand and the x-axis, bounded by x = 4 and x = 5. This region can be divided into two parts:
a rectangle with a width of 1 and a height of 3, and a triangle with a base of 1 and a height of 2.
The area of the rectangle is 1 × 3 = 3, and the area of the triangle is (1/2) × 1 ×2 = 1.
Therefore, the total area of the region is 3 + 1 = 4, which is the value of the integral.
The units of the integral are square units since we are finding the area of a region. Thus, the integral is equal to 4 square units.
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