Question 1

If the infinite curve $y=e^{x}, x \geq 0$ , is rotated about the x-axis , find the area of the resulting surface.

Accepted Answer

A)  $\frac{\pi}{6}[\sqrt{2}+\ln (1+\sqrt{2})]$ 
B)  $\frac{\pi}{6}[\sqrt{3}+\ln (1+\sqrt{3})]$ 
C)  $\frac{\pi}{3}[\sqrt{2}+\ln (1+\sqrt{2})]$ 
D)  $\pi[\sqrt{2}+\ln (1+\sqrt{2})]$ 
E)  $\frac{\pi}{4}[\sqrt{2}+\ln (1+\sqrt{2})]$ 
A)  $\frac{\pi}{6}[\sqrt{2}+\ln (1+\sqrt{2})]$ 
B)  $\frac{\pi}{6}[\sqrt{3}+\ln (1+\sqrt{3})]$ 
C)  $\frac{\pi}{3}[\sqrt{2}+\ln (1+\sqrt{2})]$ 
D)  $\pi[\sqrt{2}+\ln (1+\sqrt{2})]$ 
E)  $\frac{\pi}{4}[\sqrt{2}+\ln (1+\sqrt{2})]$

Question 2

Suppose the average waiting time for a customer's call to be answered by a company representative (modeled by exponentially decreasing probability density functions) is $20$ minutes. Find the median waiting time.

Accepted Answer

A)  $16.86$ minutes 
B)  $17.86$ minutes 
C)  $13.86$ minutes 
D)  $14.86$ minutes 
E)  $15.86$ minutes 
A)  $16.86$ minutes 
B)  $17.86$ minutes 
C)  $13.86$ minutes 
D)  $14.86$ minutes 
E)  $15.86$ minutes

Question 3

Set up, but do not evaluate, an integral that represents the length of the curve. $y=7^{x}, 0 \leq x \leq 5$

Accepted Answer

The answer of Set up, but do not evaluate, an...

Question 4

Find the arc length of the graph of the given equation from P to Q.
y = - 3x + 2; P (3, -7), Q (6, -16)

Accepted Answer

The answer of Find the arc length of the graph...

Question 5

The standard deviation for a random variable with probability density function f and mean µ is defined $ \sigma=\left[\int_{-\infty}^{\infty}(x-\mu)^{2} f(x) d x\right]^{1 / 2} $ . Find the standard deviation for an exponential density function with mean $10$ .

Accepted Answer

A)  $10$ 
B)  $2.5$ 
C)  2 
D)  $3.3$ 
E)  5 
A)  $10$ 
B)  $2.5$ 
C)  2 
D)  $3.3$ 
E)  5

Question 6

Find the area of the surface obtained by revolving the given curve about the x-axis. $y=\frac{e^{x}+e^{-x}}{2}$ on $[0, \ln 5]$

Accepted Answer

A) 
$\frac{156}{25} \pi$ 
B) $\frac{\pi}{5}\left(\frac{156}{25}+\ln 5\right)$ 
C)  $\pi\left(\frac{156}{25}+\ln 5\right)$ 
D) $\pi \ln 5$ 
A) 
$\frac{156}{25} \pi$ 
B) $\frac{\pi}{5}\left(\frac{156}{25}+\ln 5\right)$ 
C)  $\pi\left(\frac{156}{25}+\ln 5\right)$ 
D) $\pi \ln 5$

Question 7

Find the length of the curve. $y=2 \ln \left(\sin \frac{x}{2}\right), \frac{\pi}{5} \leq x \leq \pi$

Accepted Answer

A)  $\ln (2+\sqrt{5})$ 
B)  $\ln (\sqrt{5})$ 
C)  $\ln (5)$ 
D)  $\ln (2 \sqrt{5})$ 
E)  None of these 
A)  $\ln (2+\sqrt{5})$ 
B)  $\ln (\sqrt{5})$ 
C)  $\ln (5)$ 
D)  $\ln (2 \sqrt{5})$ 
E)  None of these

Question 8

Find the length of the line segment joining the two given points by finding the equation of the line using Equation (2). Then check your answer by using the distance formula. (0, 0), (1, 8)

Accepted Answer

A)  $\sqrt{62}$ 
B)  9 
C)  $\sqrt{65}$ 
D)  3 
A)  $\sqrt{62}$ 
B)  9 
C)  $\sqrt{65}$ 
D)  3

Question 9

Let the function whose graph is shown be a probability density function. Calculate the mean.

Accepted Answer

The answer of Let the function whose graph is shown...

Question 10

The marginal revenue from producing x units of a certain product is $100+x-0.001 x^{2}+0.00003 x^{3}$ (in dollars per unit). Find the increase in revenue if the production level is raised from 1,100 units to $1700$ units.

Accepted Answer

A)  $17765250 
B)  $ $51366000$ 
C)  $26866667 
D)  $37974583 
E)  $51367000 
A)  $17765250 
B)  $ $51366000$ 
C)  $26866667 
D)  $37974583 
E)  $51367000

Question 11

Find the area of the surface obtained by revolving the graph of y = $\sqrt{4-x^{2}}$ on [0, 1] about the x-axis.

Accepted Answer

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

Question 12

The masses $m_{i}$ are located at the point $P_{i}$ . Find the moments $M_{x}$ and $M_{y}$ and the center of mass of the system. $m_{1}=3, m_{2}=7, m_{3}=113$ ; $p_{1}(1,5), P_{2}(3,-2), p_{3}(-2,-1)$

Accepted Answer

A)  $M_{x}=22, M_{y}=50,\left(\frac{50}{23}, \frac{22}{23}\right)$ 
B)  $M_{x}=50, M_{y}=22,\left(\frac{50}{23}, \frac{22}{23}\right)$ 
C)  $M_{x}=-50, M_{y}=22,\left(\frac{50}{23},-\frac{22}{23}\right)$ 
D)  $M_{x}=22, M_{y}=50,\left(\frac{22}{23}, \frac{50}{23}\right)$ 
E)  $M_{x}=50, M_{y}=-22,\left(-\frac{50}{23}, \frac{22}{23}\right)$ 
A)  $M_{x}=22, M_{y}=50,\left(\frac{50}{23}, \frac{22}{23}\right)$ 
B)  $M_{x}=50, M_{y}=22,\left(\frac{50}{23}, \frac{22}{23}\right)$ 
C)  $M_{x}=-50, M_{y}=22,\left(\frac{50}{23},-\frac{22}{23}\right)$ 
D)  $M_{x}=22, M_{y}=50,\left(\frac{22}{23}, \frac{50}{23}\right)$ 
E)  $M_{x}=50, M_{y}=-22,\left(-\frac{50}{23}, \frac{22}{23}\right)$

Question 13

The manager of a fast-food restaurant determines that the average time that her customers wait for service is 2 minutes.
The manager wants to advertise that anybody who isn't served within a certain number of minutes gets a free hamburger. But she doesn't want to give away free hamburgers to more than $3$ % of her customers. What value of x must she use in the advertisement "if you aren't served within x minutes, you get a free hamburger"?

Accepted Answer

The answer of The manager of a fast-food restaurant determines...

Question 14

Use the arc length formula to find the length of the curve. $y=5-3 x,-5 \leq x \leq-5$

Accepted Answer

The answer of Use the arc length formula to find...

Question 15

Find the area of the surface obtained by rotating the curve about the x-axis. $x=\frac{1}{3}\left(y^{2}+2\right)^{\frac{3}{2}}, 1 \leq y \leq 4$

Accepted Answer

A)  $\frac{285}{2}$ $\pi$ 
B)  $283$ $\pi$ 
C)  $281$ $\pi$ 
D)  $\frac{287 \pi}{2}$ 
E)  $\frac{289 \pi}{2}$ 
A)  $\frac{285}{2}$ $\pi$ 
B)  $283$ $\pi$ 
C)  $281$ $\pi$ 
D)  $\frac{287 \pi}{2}$ 
E)  $\frac{289 \pi}{2}$

Question 16

Find the volume obtained when the circle of radius 2 with center ( 2 , 0) is rotated about the y-axis.

Accepted Answer

A)  $\frac{32}{3}$ $\pi$ 
B)  $32$ $\pi$ 
C)  $2 \pi$ 
D)  $\frac{3}{32} \pi$ 
E)  $\pi$ 
A)  $\frac{32}{3}$ $\pi$ 
B)  $32$ $\pi$ 
C)  $2 \pi$ 
D)  $\frac{3}{32} \pi$ 
E)  $\pi$

Question 17

A rectangular tank has width 4 ft, height 4 ft, and length 7 ft. It is filled with equal volumes of water and oil. The oil has a weight density of 50 lb/ft3 and floats on the water. Find the force exerted by the mixture on one end of the tank. (The weight density of water is 62.4 lb/ft3.)

Accepted Answer

A)  1897.6 lb 
B)  1699.2 lb 
C)  899.2 lb 
D)  1798.4 lb 
A)  1897.6 lb 
B)  1699.2 lb 
C)  899.2 lb 
D)  1798.4 lb

Question 18

Set up, but do not evaluate, an integral for the area of the surface obtained by rotating the curve $y=\ln x^{7}$ about the x-axis on the interval $1 \leq x \leq 7$ .

Accepted Answer

A)  $\int_{7}^{1} 2 \pi \sqrt{1+\left(\frac{7}{x}\right)^{2}} d x$ 
B)  $\int_{7}^{1} 2 \pi x \ln (x) \sqrt{1+\left(\frac{7}{x}\right)^{2}} d x$ 
C)  $\int_{1}^{7} 2 \pi(7 x) \sqrt{1+\left(\frac{7}{x}\right)^{2}} d x$ 
D)  $\int_{7}^{1} 2 \pi(7 x) \sqrt{1+\left(\frac{7}{x}\right)^{2}} d x$ 
E)  $\int_{1}^{7} 2 \pi(7 \ln (x)) \sqrt{1+\left(\frac{7}{x}\right)^{2}} d x$ 
A)  $\int_{7}^{1} 2 \pi \sqrt{1+\left(\frac{7}{x}\right)^{2}} d x$ 
B)  $\int_{7}^{1} 2 \pi x \ln (x) \sqrt{1+\left(\frac{7}{x}\right)^{2}} d x$ 
C)  $\int_{1}^{7} 2 \pi(7 x) \sqrt{1+\left(\frac{7}{x}\right)^{2}} d x$ 
D)  $\int_{7}^{1} 2 \pi(7 x) \sqrt{1+\left(\frac{7}{x}\right)^{2}} d x$ 
E)  $\int_{1}^{7} 2 \pi(7 \ln (x)) \sqrt{1+\left(\frac{7}{x}\right)^{2}} d x$

Question 19

Find the coordinates of the centroid for the region bounded by the curves $y=2 x^{2}$ , x = 0,
and y = 72 .

Accepted Answer

The answer of Find the coordinates of the centroid for...

Question 20

Find the center of mass of a lamina in the shape of a quarter-circle with radius 9 with density $\rho$ = 3 .

Accepted Answer

The answer of Find the center of mass of a...

If the infinite curve $y=e^{x}, x \geq 0$ , is rotated about the x-axis , find the area of the resulting surface.

Suppose the average waiting time for a customer's call to be answered by a company representative (modeled by exponentially decreasing probability density functions) is $20$ minutes. Find the median waiting time.

Set up, but do not evaluate, an integral that represents the length of the curve. $y=7^{x}, 0 \leq x \leq 5$

Find the arc length of the graph of the given equation from P to Q. y = - 3x + 2; P (3, -7), Q (6, -16)

The standard deviation for a random variable with probability density function f and mean µ is defined $\sigma=\left[\int_{-\infty}^{\infty}(x-\mu)^{2} f(x) d x\right]^{1 / 2}$ . Find the standard deviation for an exponential density function with mean $10$ .

Find the area of the surface obtained by revolving the given curve about the x-axis. $y=\frac{e^{x}+e^{-x}}{2}$ on $[0, \ln 5]$

Find the length of the curve. $y=2 \ln \left(\sin \frac{x}{2}\right), \frac{\pi}{5} \leq x \leq \pi$

Find the length of the line segment joining the two given points by finding the equation of the line using Equation (2). Then check your answer by using the distance formula. (0, 0), (1, 8)

Let the function whose graph is shown be a probability density function. Calculate the mean.

The marginal revenue from producing x units of a certain product is $100+x-0.001 x^{2}+0.00003 x^{3}$ (in dollars per unit). Find the increase in revenue if the production level is raised from 1,100 units to $1700$ units.

Find the area of the surface obtained by revolving the graph of y = $\sqrt{4-x^{2}}$ on [0, 1] about the x-axis.

The masses $m_{i}$ are located at the point $P_{i}$ . Find the moments $M_{x}$ and $M_{y}$ and the center of mass of the system. $m_{1}=3, m_{2}=7, m_{3}=113$ ; $p_{1}(1,5), P_{2}(3,-2), p_{3}(-2,-1)$

Use the arc length formula to find the length of the curve. $y=5-3 x,-5 \leq x \leq-5$

Find the area of the surface obtained by rotating the curve about the x-axis. $x=\frac{1}{3}\left(y^{2}+2\right)^{\frac{3}{2}}, 1 \leq y \leq 4$

Find the volume obtained when the circle of radius 2 with center ( 2 , 0) is rotated about the y-axis.

A rectangular tank has width 4 ft, height 4 ft, and length 7 ft. It is filled with equal volumes of water and oil. The oil has a weight density of 50 lb/ft³ and floats on the water. Find the force exerted by the mixture on one end of the tank. (The weight density of water is 62.4 lb/ft³.)

Set up, but do not evaluate, an integral for the area of the surface obtained by rotating the curve $y=\ln x^{7}$ about the x-axis on the interval $1 \leq x \leq 7$ .

Find the coordinates of the centroid for the region bounded by the curves $y=2 x^{2}$ , x = 0, and y = 72 .

Find the center of mass of a lamina in the shape of a quarter-circle with radius 9 with density $\rho$ = 3 . $Find the center of mass of a lamina in the shape of a quarter-circle with radius 9 with density \rho = 3 .$

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

If the infinite curve y=ex,x≥0y=e^{x}, x \geq 0y=ex,x≥0 , is rotated about the x-axis , find the area of the resulting surface.

Suppose the average waiting time for a customer's call to be answered by a company representative (modeled by exponentially decreasing probability density functions) is 202020 minutes. Find the median waiting time.

Set up, but do not evaluate, an integral that represents the length of the curve. y=7x,0≤x≤5y=7^{x}, 0 \leq x \leq 5y=7x,0≤x≤5

Find the arc length of the graph of the given equation from P to Q. y = - 3x + 2; P (3, -7), Q (6, -16)

Find the area of the surface obtained by revolving the given curve about the x-axis. y=ex+e−x2y=\frac{e^{x}+e^{-x}}{2}y=2ex+e−x​ on [0,ln⁡5][0, \ln 5][0,ln5]

Find the length of the curve. y=2ln⁡(sin⁡x2),π5≤x≤πy=2 \ln \left(\sin \frac{x}{2}\right), \frac{\pi}{5} \leq x \leq \piy=2ln(sin2x​),5π​≤x≤π

Find the length of the line segment joining the two given points by finding the equation of the line using Equation (2). Then check your answer by using the distance formula. (0, 0), (1, 8)

Let the function whose graph is shown be a probability density function. Calculate the mean.

The marginal revenue from producing x units of a certain product is 100+x−0.001x2+0.00003x3100+x-0.001 x^{2}+0.00003 x^{3}100+x−0.001x2+0.00003x3 (in dollars per unit). Find the increase in revenue if the production level is raised from 1,100 units to 170017001700 units.

Find the area of the surface obtained by revolving the graph of y = 4−x2\sqrt{4-x^{2}}4−x2​ on [0, 1] about the x-axis.

Use the arc length formula to find the length of the curve. y=5−3x,−5≤x≤−5y=5-3 x,-5 \leq x \leq-5y=5−3x,−5≤x≤−5

Find the area of the surface obtained by rotating the curve about the x-axis. x=13(y2+2)32,1≤y≤4x=\frac{1}{3}\left(y^{2}+2\right)^{\frac{3}{2}}, 1 \leq y \leq 4x=31​(y2+2)23​,1≤y≤4

Find the volume obtained when the circle of radius 2 with center ( 2 , 0) is rotated about the y-axis.

A rectangular tank has width 4 ft, height 4 ft, and length 7 ft. It is filled with equal volumes of water and oil. The oil has a weight density of 50 lb/ft3 and floats on the water. Find the force exerted by the mixture on one end of the tank. (The weight density of water is 62.4 lb/ft3.)

Set up, but do not evaluate, an integral for the area of the surface obtained by rotating the curve y=ln⁡x7y=\ln x^{7}y=lnx7 about the x-axis on the interval 1≤x≤71 \leq x \leq 71≤x≤7 .

Find the coordinates of the centroid for the region bounded by the curves y=2x2y=2 x^{2}y=2x2 , x = 0, and y = 72 .

Find the center of mass of a lamina in the shape of a quarter-circle with radius 9 with density ρ\rhoρ = 3 .

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

If the infinite curve $y=e^{x}, x \geq 0$ , is rotated about the x-axis , find the area of the resulting surface.

Suppose the average waiting time for a customer's call to be answered by a company representative (modeled by exponentially decreasing probability density functions) is $20$ minutes. Find the median waiting time.

Set up, but do not evaluate, an integral that represents the length of the curve. $y=7^{x}, 0 \leq x \leq 5$

Find the area of the surface obtained by revolving the given curve about the x-axis. $y=\frac{e^{x}+e^{-x}}{2}$ on $[0, \ln 5]$

Find the length of the curve. $y=2 \ln \left(\sin \frac{x}{2}\right), \frac{\pi}{5} \leq x \leq \pi$

The marginal revenue from producing x units of a certain product is $100+x-0.001 x^{2}+0.00003 x^{3}$ (in dollars per unit). Find the increase in revenue if the production level is raised from 1,100 units to $1700$ units.

Find the area of the surface obtained by revolving the graph of y = $\sqrt{4-x^{2}}$ on [0, 1] about the x-axis.

Use the arc length formula to find the length of the curve. $y=5-3 x,-5 \leq x \leq-5$

Find the area of the surface obtained by rotating the curve about the x-axis. $x=\frac{1}{3}\left(y^{2}+2\right)^{\frac{3}{2}}, 1 \leq y \leq 4$

A rectangular tank has width 4 ft, height 4 ft, and length 7 ft. It is filled with equal volumes of water and oil. The oil has a weight density of 50 lb/ft³ and floats on the water. Find the force exerted by the mixture on one end of the tank. (The weight density of water is 62.4 lb/ft³.)

Set up, but do not evaluate, an integral for the area of the surface obtained by rotating the curve $y=\ln x^{7}$ about the x-axis on the interval $1 \leq x \leq 7$ .

Find the coordinates of the centroid for the region bounded by the curves $y=2 x^{2}$ , x = 0, and y = 72 .

Find the center of mass of a lamina in the shape of a quarter-circle with radius 9 with density $\rho$ = 3 . $Find the center of mass of a lamina in the shape of a quarter-circle with radius 9 with density \rho = 3 .$