Question 1

Write an integral giving the area of the surface obtained by revolving the curve about the x-axis. (Do not evaluate the integral.) y = $\frac{4}{x}$ on [3, 6]

Accepted Answer

A)  (F1F1F1S1F1F1F10  ) 
B)  8F1F1F1S1F1F1F10 $\int_{3}^{6} x^{3} \sqrt{x^{4}+16} d x$ 
C)  8F1F1F1S1F1F1F10 $\int_{3}^{6} \frac{\sqrt{x^{4}+16}}{x^{3}} d x$ 
D)  (F1F1F1S1F1F1F10 $\int_{3}^{6} \frac{4}{x}\left(1+\left(-\frac{4}{x^{2}}\right)^{2}\right) d x$ ) 
A)  (F1F1F1S1F1F1F10  ) 
B)  8F1F1F1S1F1F1F10 $\int_{3}^{6} x^{3} \sqrt{x^{4}+16} d x$ 
C)  8F1F1F1S1F1F1F10 $\int_{3}^{6} \frac{\sqrt{x^{4}+16}}{x^{3}} d x$ 
D)  (F1F1F1S1F1F1F10 $\int_{3}^{6} \frac{4}{x}\left(1+\left(-\frac{4}{x^{2}}\right)^{2}\right) d x$ ) C

Question 2

Set up, but do not evaluate, an integral for the length of the curve. $y=2 e^{x} \sin x, \quad \quad 0 \leq x \leq \frac{9 \pi}{2}$

Accepted Answer

A)  $L=\int_{0}^{9 x / 2} \sqrt{1-4 e^{2 x}(1+\sin 2 x)} d x$ 
B)  $L=\int_{0}^{9 x / 2} \sqrt{1+4 e^{2 x}(1-\sin x)} d x$ 
C)  $L=\int_{0}^{9 x / 2} \sqrt{1-4 e^{2 x}(1-\sin 2 x)} d x$ 
D)  $L=\int_{0}^{9 x / 2} \sqrt{1+4 e^{2 x}(1-\sin 2 x)} d x$ 
E)  $L=\int_{0}^{9 x / 2} \sqrt{1+4 e^{2 x}(1+\sin 2 x)} d x$ 
A)  $L=\int_{0}^{9 x / 2} \sqrt{1-4 e^{2 x}(1+\sin 2 x)} d x$ 
B)  $L=\int_{0}^{9 x / 2} \sqrt{1+4 e^{2 x}(1-\sin x)} d x$ 
C)  $L=\int_{0}^{9 x / 2} \sqrt{1-4 e^{2 x}(1-\sin 2 x)} d x$ 
D)  $L=\int_{0}^{9 x / 2} \sqrt{1+4 e^{2 x}(1-\sin 2 x)} d x$ 
E)  $L=\int_{0}^{9 x / 2} \sqrt{1+4 e^{2 x}(1+\sin 2 x)} d x$ E

Question 3

A trough has vertical ends that are equilateral triangles with sides of length 2 ft. If the trough is filled with water to a depth of 1 ft, find the force exerted by the water on one end of the trough. Round to one decimal place. (The weight density of water is 62.4 lb/ft3.)

Accepted Answer

A)  31.2 lb 
B)  12.0 lb 
C)  62.4 lb 
D)  6.0 lb 
A)  31.2 lb 
B)  12.0 lb 
C)  62.4 lb 
D)  6.0 lb B

Question 4

You are given the shape of the vertical ends of a trough that is completely filled with water. Find the force exerted by the water on one end of the trough. (The weight density of water is 62.4 lb/ft3.)

Accepted Answer

A)  2662.4 lb 
B)  1331.2 lb 
C)  665.6 lb 
D)  332.8 lb 
A)  2662.4 lb 
B)  1331.2 lb 
C)  665.6 lb 
D)  332.8 lb

Question 5

For the following exercise, (a) plot the graph of the function f, (b) write an integral giving the arc length of the graph of the function over the indicated interval, and (c) find the arc length of the curve accurate to two decimal places. $f(x)$ = x - 3 $\sqrt{x}$ ; [0, 3]

Accepted Answer

The answer of For the following exercise, (a) plot the...

Question 6

Find the centroid of the region bounded by the given curves. $y=4 \sin 5 x, y=0, x=0, x=\frac{\pi}{5}$

Accepted Answer

A)  $(0.3146,6.2832)$ 
B)  $(1.2565,6.2832)$ 
C)  $(-1.2565,6.2832)$ 
D)  $(1.2565,0.3146)$ 
E)  $(-0.3146,0.7854)$ 
A)  $(0.3146,6.2832)$ 
B)  $(1.2565,6.2832)$ 
C)  $(-1.2565,6.2832)$ 
D)  $(1.2565,0.3146)$ 
E)  $(-0.3146,0.7854)$

Question 7

Find the area of the surface obtained by rotating the curve about the $X$ -axis. $y=x^{3}, 0 \leq x \leq 2$
 on [0, 2]

Accepted Answer

The answer of Find the area of the surface obtained...

Question 8

The manager of a fast-food restaurant determines that the average time that her customers wait for service is 2 minutes. Find the probability that a customer is served within the first 2 minutes.

Accepted Answer

A)  $1.6321$ 
B)  $0.6321$ 
C)  $4.6321$ 
D)  $3.6321$ 
E)  $2.6321$ 
A)  $1.6321$ 
B)  $0.6321$ 
C)  $4.6321$ 
D)  $3.6321$ 
E)  $2.6321$

Question 9

Find the area of the surface obtained by rotating the circle $x^{2}+y^{2}=5^{2}$ about the line $y=5$ .

Accepted Answer

A)  $25 \pi^{2}$ 
B)  $75 \pi^{2}$ 
C)  $175 \pi^{2}$ 
D)  $50 \pi^{2}$ 
E)  $100 \pi^{2}$ 
A)  $25 \pi^{2}$ 
B)  $75 \pi^{2}$ 
C)  $175 \pi^{2}$ 
D)  $50 \pi^{2}$ 
E)  $100 \pi^{2}$

Question 10

Use Simpson's Rule with n = 6 to estimate the length of the curve $y=3 \sin x$ , $0 \leq x \leq \pi$ . Round your answer to six decimal places.

Accepted Answer

A)  6.987208 
B)  6.947368 
C)  6.972089 
D)  6.947582 
E)  None of these 
A)  6.987208 
B)  6.947368 
C)  6.972089 
D)  6.947582 
E)  None of these

Question 11

A demand curve is given by $p=\frac{300}{x+8}$ .
Find the consumer surplus when the selling price is $ $20$ .

Accepted Answer

The answer of A demand curve is given by $p=\frac{300}{x+8}$...

Question 12

You are given the shape of the vertical ends of a trough that is completely filled with water. Find the force exerted by the water on one end of the trough. (The weight density of water is 62.4 lb/ft3.)

Accepted Answer

Question 13

You are given the shape of the vertical ends of a trough that is completely filled with water. Find the force exerted by the water on one end of the trough. (The weight density of water is 62.4 lb/ft3.)

Accepted Answer

A)  104 lb 
B)  1040 lb 
C)  208 lb 
D)  520 lb 
A)  104 lb 
B)  1040 lb 
C)  208 lb 
D)  520 lb

Question 14

Find the length of the curve. $y=\frac{1}{6}\left(x^{2}+4\right)^{3 / 2}, 0 \leq x \leq 3$

Accepted Answer

A)  $6.5000$ 
B)  $7.5000$ 
C)  $8.5000$ 
D)  $5.5000$ 
E)  $4.5000$ 
A)  $6.5000$ 
B)  $7.5000$ 
C)  $8.5000$ 
D)  $5.5000$ 
E)  $4.5000$

Question 15

The demand function for a commodity is given by $p=2000-0.1 x-0.01 x^{2}$ . Find the consumer surplus when the sales level is $125$ .

Accepted Answer

A)  $10458 
B)  $4458 
C)  $ $6458$ 
D)  $9458 
E)  $8458 
A)  $10458 
B)  $4458 
C)  $ $6458$ 
D)  $9458 
E)  $8458

Question 16

Use differentials to approximate the arc length of the graph of the equation from P to Q. Round answer to four decimal places. $y=x^{2}+3$ P (3, 12), Q (3.2, 13.24)

Accepted Answer

A)  6.0828 
B)  3.6497 
C)  1.2166 
D)  2.4331 
A)  6.0828 
B)  3.6497 
C)  1.2166 
D)  2.4331

Question 17

A vertical plate is submerged in water (the surface of the water coincides with the x-axis). Find the force exerted by the water on the plate. (The weight density of water is 62.4 lb/ft3.)   (ft)

Accepted Answer

Question 18

Find the centroid of the region shown, not by integration, but by locating the centroids of the
rectangles and triangles and using additivity of moments.

Accepted Answer

The answer of Find the centroid of the region shown,...

Question 19

Calculate the center of mass of the lamina with density $\rho$ = 6 .

Accepted Answer

The answer of Calculate the center of mass of the...

Question 20

Find the length of the curve for the interval $1 \leq x \leq 4$ . $y=\int_{1}^{x} \sqrt{t^{4}-1} d t$

Accepted Answer

A)  $L=15.7500$ 
B)  $L=10.5000$ 
C)  $L=12.6000$ 
D)  $L=21.0000$ 
E)  $L=0.0476$ 
A)  $L=15.7500$ 
B)  $L=10.5000$ 
C)  $L=12.6000$ 
D)  $L=21.0000$ 
E)  $L=0.0476$

Write an integral giving the area of the surface obtained by revolving the curve about the x-axis. (Do not evaluate the integral.) y = $\frac{4}{x}$ on [3, 6]

Set up, but do not evaluate, an integral for the length of the curve. $y=2 e^{x} \sin x, \quad \quad 0 \leq x \leq \frac{9 \pi}{2}$

A trough has vertical ends that are equilateral triangles with sides of length 2 ft. If the trough is filled with water to a depth of 1 ft, find the force exerted by the water on one end of the trough. Round to one decimal place. (The weight density of water is 62.4 lb/ft³.)

You are given the shape of the vertical ends of a trough that is completely filled with water. Find the force exerted by the water on one end of the trough. (The weight density of water is 62.4 lb/ft³.)

For the following exercise, (a) plot the graph of the function f, (b) write an integral giving the arc length of the graph of the function over the indicated interval, and (c) find the arc length of the curve accurate to two decimal places. $f(x)$ = x - 3 $\sqrt{x}$ ; [0, 3]

Find the centroid of the region bounded by the given curves. $y=4 \sin 5 x, y=0, x=0, x=\frac{\pi}{5}$

Find the area of the surface obtained by rotating the curve about the $X$ -axis. $y=x^{3}, 0 \leq x \leq 2$ on [0, 2]

The manager of a fast-food restaurant determines that the average time that her customers wait for service is 2 minutes. Find the probability that a customer is served within the first 2 minutes.

Find the area of the surface obtained by rotating the circle $x^{2}+y^{2}=5^{2}$ about the line $y=5$ .

Use Simpson's Rule with n = 6 to estimate the length of the curve $y=3 \sin x$ , $0 \leq x \leq \pi$ . Round your answer to six decimal places.

A demand curve is given by $p=\frac{300}{x+8}$ . Find the consumer surplus when the selling price is $ $20$ .

You are given the shape of the vertical ends of a trough that is completely filled with water. Find the force exerted by the water on one end of the trough. (The weight density of water is 62.4 lb/ft³.)

You are given the shape of the vertical ends of a trough that is completely filled with water. Find the force exerted by the water on one end of the trough. (The weight density of water is 62.4 lb/ft³.)

Find the length of the curve. $y=\frac{1}{6}\left(x^{2}+4\right)^{3 / 2}, 0 \leq x \leq 3$

The demand function for a commodity is given by $p=2000-0.1 x-0.01 x^{2}$ . Find the consumer surplus when the sales level is $125$ .

Use differentials to approximate the arc length of the graph of the equation from P to Q. Round answer to four decimal places. $y=x^{2}+3$ P (3, 12), Q (3.2, 13.24)

A vertical plate is submerged in water (the surface of the water coincides with the x-axis). Find the force exerted by the water on the plate. (The weight density of water is 62.4 lb/ft³.) (ft)

Find the centroid of the region shown, not by integration, but by locating the centroids of the rectangles and triangles and using additivity of moments.

Calculate the center of mass of the lamina with density $\rho$ = 6 . $Calculate the center of mass of the lamina with density \rho = 6 .$

Find the length of the curve for the interval $1 \leq x \leq 4$ . $y=\int_{1}^{x} \sqrt{t^{4}-1} d t$

Functions and Limits

Derivatives

Applications of Differentiation

Integrals

Applications of Integration

Inverse Functions

Techniques of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Final Exam

Filters

Exam 8: Further Applications of Integration

Write an integral giving the area of the surface obtained by revolving the curve about the x-axis. (Do not evaluate the integral.) y = 4x\frac{4}{x}x4​ on [3, 6]

Set up, but do not evaluate, an integral for the length of the curve. y=2exsin⁡x,0≤x≤9π2y=2 e^{x} \sin x, \quad \quad 0 \leq x \leq \frac{9 \pi}{2}y=2exsinx,0≤x≤29π​

A trough has vertical ends that are equilateral triangles with sides of length 2 ft. If the trough is filled with water to a depth of 1 ft, find the force exerted by the water on one end of the trough. Round to one decimal place. (The weight density of water is 62.4 lb/ft3.)

You are given the shape of the vertical ends of a trough that is completely filled with water. Find the force exerted by the water on one end of the trough. (The weight density of water is 62.4 lb/ft3.)

For the following exercise, (a) plot the graph of the function f, (b) write an integral giving the arc length of the graph of the function over the indicated interval, and (c) find the arc length of the curve accurate to two decimal places. f(x)f(x)f(x) = x - 3 x\sqrt{x}x​ ; [0, 3]

Find the centroid of the region bounded by the given curves. y=4sin⁡5x,y=0,x=0,x=π5y=4 \sin 5 x, y=0, x=0, x=\frac{\pi}{5}y=4sin5x,y=0,x=0,x=5π​

Find the area of the surface obtained by rotating the curve about the XXX -axis. y=x3,0≤x≤2y=x^{3}, 0 \leq x \leq 2y=x3,0≤x≤2 on [0, 2]

The manager of a fast-food restaurant determines that the average time that her customers wait for service is 2 minutes. Find the probability that a customer is served within the first 2 minutes.

Find the area of the surface obtained by rotating the circle x2+y2=52x^{2}+y^{2}=5^{2}x2+y2=52 about the line y=5y=5y=5 .

Use Simpson's Rule with n = 6 to estimate the length of the curve y=3sin⁡xy=3 \sin xy=3sinx , 0≤x≤π0 \leq x \leq \pi0≤x≤π . Round your answer to six decimal places.

A demand curve is given by p=300x+8p=\frac{300}{x+8}p=x+8300​ . Find the consumer surplus when the selling price is $ 202020 .

You are given the shape of the vertical ends of a trough that is completely filled with water. Find the force exerted by the water on one end of the trough. (The weight density of water is 62.4 lb/ft3.)

You are given the shape of the vertical ends of a trough that is completely filled with water. Find the force exerted by the water on one end of the trough. (The weight density of water is 62.4 lb/ft3.)

Find the length of the curve. y=16(x2+4)3/2,0≤x≤3y=\frac{1}{6}\left(x^{2}+4\right)^{3 / 2}, 0 \leq x \leq 3y=61​(x2+4)3/2,0≤x≤3

The demand function for a commodity is given by p=2000−0.1x−0.01x2p=2000-0.1 x-0.01 x^{2}p=2000−0.1x−0.01x2 . Find the consumer surplus when the sales level is 125125125 .

Use differentials to approximate the arc length of the graph of the equation from P to Q. Round answer to four decimal places. y=x2+3y=x^{2}+3y=x2+3 P (3, 12), Q (3.2, 13.24)

A vertical plate is submerged in water (the surface of the water coincides with the x-axis). Find the force exerted by the water on the plate. (The weight density of water is 62.4 lb/ft3.) (ft)

Find the centroid of the region shown, not by integration, but by locating the centroids of the rectangles and triangles and using additivity of moments.

Calculate the center of mass of the lamina with density ρ\rhoρ = 6 .

Find the length of the curve for the interval 1≤x≤41 \leq x \leq 41≤x≤4 . y=∫1xt4−1dty=\int_{1}^{x} \sqrt{t^{4}-1} d ty=∫1x​t4−1​dt

Functions and Limits

Derivatives

Applications of Differentiation

Integrals

Applications of Integration

Inverse Functions

Techniques of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Final Exam

Filters

Write an integral giving the area of the surface obtained by revolving the curve about the x-axis. (Do not evaluate the integral.) y = $\frac{4}{x}$ on [3, 6]

Set up, but do not evaluate, an integral for the length of the curve. $y=2 e^{x} \sin x, \quad \quad 0 \leq x \leq \frac{9 \pi}{2}$

A trough has vertical ends that are equilateral triangles with sides of length 2 ft. If the trough is filled with water to a depth of 1 ft, find the force exerted by the water on one end of the trough. Round to one decimal place. (The weight density of water is 62.4 lb/ft³.)

You are given the shape of the vertical ends of a trough that is completely filled with water. Find the force exerted by the water on one end of the trough. (The weight density of water is 62.4 lb/ft³.)

For the following exercise, (a) plot the graph of the function f, (b) write an integral giving the arc length of the graph of the function over the indicated interval, and (c) find the arc length of the curve accurate to two decimal places. $f(x)$ = x - 3 $\sqrt{x}$ ; [0, 3]

Find the centroid of the region bounded by the given curves. $y=4 \sin 5 x, y=0, x=0, x=\frac{\pi}{5}$

Find the area of the surface obtained by rotating the curve about the $X$ -axis. $y=x^{3}, 0 \leq x \leq 2$ on [0, 2]

Find the area of the surface obtained by rotating the circle $x^{2}+y^{2}=5^{2}$ about the line $y=5$ .

Use Simpson's Rule with n = 6 to estimate the length of the curve $y=3 \sin x$ , $0 \leq x \leq \pi$ . Round your answer to six decimal places.

A demand curve is given by $p=\frac{300}{x+8}$ . Find the consumer surplus when the selling price is $ $20$ .

You are given the shape of the vertical ends of a trough that is completely filled with water. Find the force exerted by the water on one end of the trough. (The weight density of water is 62.4 lb/ft³.)

You are given the shape of the vertical ends of a trough that is completely filled with water. Find the force exerted by the water on one end of the trough. (The weight density of water is 62.4 lb/ft³.)

Find the length of the curve. $y=\frac{1}{6}\left(x^{2}+4\right)^{3 / 2}, 0 \leq x \leq 3$

The demand function for a commodity is given by $p=2000-0.1 x-0.01 x^{2}$ . Find the consumer surplus when the sales level is $125$ .

Use differentials to approximate the arc length of the graph of the equation from P to Q. Round answer to four decimal places. $y=x^{2}+3$ P (3, 12), Q (3.2, 13.24)

A vertical plate is submerged in water (the surface of the water coincides with the x-axis). Find the force exerted by the water on the plate. (The weight density of water is 62.4 lb/ft³.) (ft)

Calculate the center of mass of the lamina with density $\rho$ = 6 . $Calculate the center of mass of the lamina with density \rho = 6 .$

Find the length of the curve for the interval $1 \leq x \leq 4$ . $y=\int_{1}^{x} \sqrt{t^{4}-1} d t$