Question 1

Find the centroid of the region bounded by the given curves. Select the correct answer.
$$y = 2 \sin 5 x , y = 0 , x = 0 , x = \frac { \pi } { 5 }$$

Accepted Answer

A)  $( 0.3146,1.5708 )$ 
B)  $( - 0.6283,1.5708 )$ 
C)  $( 0.2513,0.6283 )$ 
D)  $( - 0.3146,0.7854 )$ 
E)  $( 0.6283,0.3146 )$ 
A)  $( 0.3146,1.5708 )$ 
B)  $( - 0.6283,1.5708 )$ 
C)  $( 0.2513,0.6283 )$ 
D)  $( - 0.3146,0.7854 )$ 
E)  $( 0.6283,0.3146 )$

Question 2

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.
$$f ( x ) = x - 2 \sqrt { x } ; [ 0,3 ]$$

Accepted Answer

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

Question 3

Find the area of the surface obtained by rotating the circle $x ^ { 2 } + y ^ { 2 } = 7 ^ { 2 }$ about the line $y = 7$.

Accepted Answer

A)  $147 \pi ^ { 2 }$ 
B)  $49 \pi ^ { 2 }$ 
C)  $98 \pi ^ { 2 }$ 
D)  $343 \pi ^ { 2 }$ 
E)  $196 \pi ^ { 2 }$ 
A)  $147 \pi ^ { 2 }$ 
B)  $49 \pi ^ { 2 }$ 
C)  $98 \pi ^ { 2 }$ 
D)  $343 \pi ^ { 2 }$ 
E)  $196 \pi ^ { 2 }$

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 $62.4 \mathrm { lb } / \mathrm { ft } ^ { 3 }$.)

Accepted Answer

The answer of You are given the shape of the...

Question 5

Find the centroid of the region bounded by the curves.
$$y = 9 \ln 2 x , y = 0 , x = \frac { e } { 2 }$$

Accepted Answer

The answer of Find the centroid of the region bounded...

Question 6

Find the arc length of the graph of the given equation on the specified interval.
$$y = \frac { 2 } { 3 } \left( x ^ { 2 } + 1 \right) ^ { 3 / 2 } , [ 2,3 ]$$

Accepted Answer

A)  $\frac { 41 } { 3 }$ 
B)  $\frac { 41 } { 2 }$ 
C)  $\frac { 41 } { 4 }$ 
D)  $\frac { 41 } { 6 }$ 
A)  $\frac { 41 } { 3 }$ 
B)  $\frac { 41 } { 2 }$ 
C)  $\frac { 41 } { 4 }$ 
D)  $\frac { 41 } { 6 }$

Question 7

An aquarium is $3 \mathrm { ft }$ long, $2 \mathrm { ft }$ wide, and $2 \mathrm { ft }$ deep. If the aquarium is filled with water, find the force exerted by the water (a) on the bottom of the aquarium, (b) on the longer side of the aquarium, and (c) on the shorter side of the aquarium. (The weight density of water is $62.4 \mathrm { lb } / \mathrm { ft } ^ { 3 }$.)

Accepted Answer

The answer of An aquarium is \(3 \mathrm { ft...

Question 8

Find the center of mass of the lamina of the region shown if the density of the circular lamina is six times that of the rectangular lamina.

Accepted Answer

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

Question 9

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

Accepted Answer

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

Question 10

Write an integral giving the area of the surface obtained by revolving the curve about the $x$-axis. (Do not evaluate the integral.)
$$y = \frac { 2 } { x } \text { on } [ 4,5 ]$$

Accepted Answer

A)  $2 \pi \int _ { 4 } ^ { 5 } \frac { 2 } { x } \left( 1 + \left( - \frac { 2 } { x ^ { 2 } } \right) ^ { 2 } \right) d x$ 
B)  $2 \pi \int _ { 4 } ^ { 5 } \frac { \sqrt { x ^ { 4 } - 4 } } { x ^ { 3 } } d x$ 
C)  $4 \pi \int _ { 4 } ^ { 5 } x ^ { 3 } \sqrt { x ^ { 4 } + 4 } d x$ 
D) 
$$4 \pi \int _ { 4 } ^ { 5 } \frac { \sqrt { x ^ { 4 } + 4 } } { x ^ { 3 } } d x$$ 
A)  $2 \pi \int _ { 4 } ^ { 5 } \frac { 2 } { x } \left( 1 + \left( - \frac { 2 } { x ^ { 2 } } \right) ^ { 2 } \right) d x$ 
B)  $2 \pi \int _ { 4 } ^ { 5 } \frac { \sqrt { x ^ { 4 } - 4 } } { x ^ { 3 } } d x$ 
C)  $4 \pi \int _ { 4 } ^ { 5 } x ^ { 3 } \sqrt { x ^ { 4 } + 4 } d x$ 
D) 
$$4 \pi \int _ { 4 } ^ { 5 } \frac { \sqrt { x ^ { 4 } + 4 } } { x ^ { 3 } } d x$$

Question 11

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 $62.4 \mathrm { lb } / \mathrm { ft } ^ { 3 }$.)

Accepted Answer

The answer of A vertical plate is submerged in water...

Question 12

Find the centroid of the region bounded by the graphs of the given equations.
$$y = | x | \sqrt { 16 - x ^ { 2 } } , \quad y = 0 , \quad x = - 4 , \quad x = 4$$

Accepted Answer

A)  $\left( \frac { 5 } { 16 } , 0 \right)$ 
B)  $\left( \frac { 16 } { 5 } , 0 \right)$ 
C)  $\left( 0 , \frac { 5 } { 16 } \right)$ 
D)  $\left( 0 , \frac { 16 } { 5 } \right)$ 
A)  $\left( \frac { 5 } { 16 } , 0 \right)$ 
B)  $\left( \frac { 16 } { 5 } , 0 \right)$ 
C)  $\left( 0 , \frac { 5 } { 16 } \right)$ 
D)  $\left( 0 , \frac { 16 } { 5 } \right)$

Question 13

Find the centroid of the region bounded by the graphs of the given equations.
$$y = 15 - x ^ { 2 } , \quad y = 3 - x$$

Accepted Answer

The answer of Find the centroid of the region bounded...

Question 14

Dye dilution is a method of measuring cardiac output. If $A \mathrm { mg }$ of dye is used and $c ( t )$ is the concentration of the dye at time $t$, then the cardiac output over the time interval $[ 0 , T ]$ is given by
$$F = \frac { A } { \int _ { 0 } ^ { T } c ( t ) d t }$$
Find the cardiac output over the time interval $[ 0,15 ]$ if the dye dilution method is used with $11 \mathrm { mg }$ of dye and the dye concentration, in $\mathrm { mg } / \mathrm { L }$, is modeled by
$$c ( t ) = \frac { 1 } { 2 } t ( 15 - t ) , 0 \leq t \leq 15$$
where $t$ is measured in seconds.

Accepted Answer

A)  $\quad 0.031$ 
B)  $0.043$ 
C)  $0.029$ 
D)  $\quad 0.049$ 
E)  $0.039$
F) $0.022$ 
A)  $\quad 0.031$ 
B)  $0.043$ 
C)  $0.029$ 
D)  $\quad 0.049$ 
E)  $0.039$
F) $0.022$

Question 15

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 1,700 units.

Accepted Answer

A)  $\$ 37,974,583$ 
B)  $\$ 26,866,667$ 
C)  $\$ 17,765,250$ 
D)  $\$ 51,366,000$ 
E)  $\$ 51,367,000$ 
A)  $\$ 37,974,583$ 
B)  $\$ 26,866,667$ 
C)  $\$ 17,765,250$ 
D)  $\$ 51,366,000$ 
E)  $\$ 51,367,000$

Question 16

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 $62.4 \mathrm { lb } / \mathrm { ft } ^ { 3 }$.)

Accepted Answer

The answer of You are given the shape of the...

Question 17

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

Accepted Answer

A)  $15.3333$ 
B)  $14.3333$ 
C)  $14.0000$ 
D)  $14.6667$ 
E)  $13.6667$ 
A)  $15.3333$ 
B)  $14.3333$ 
C)  $14.0000$ 
D)  $14.6667$ 
E)  $13.6667$

Question 18

Find the centroid of the region bounded by the given curves.
$$y = x ^ { 3 } , x + y = 2 , x = 0$$

Accepted Answer

A)  $\frac { 29 } { 75 } , \frac { 93 } { 105 }$ 
B)  $\frac { 28 } { 75 } , \frac { 92 } { 105 }$ 
C)  $\frac { 92 } { 105 } , \frac { 9 } { 75 }$ 
D)  $\frac { 93 } { 75 } , \frac { 29 } { 105 }$ 
E)  None of these 
A)  $\frac { 29 } { 75 } , \frac { 93 } { 105 }$ 
B)  $\frac { 28 } { 75 } , \frac { 92 } { 105 }$ 
C)  $\frac { 92 } { 105 } , \frac { 9 } { 75 }$ 
D)  $\frac { 93 } { 75 } , \frac { 29 } { 105 }$ 
E)  None of these

Question 19

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

Accepted Answer

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

Question 20

The standard deviation for a random variable with probability density function $f$ and mean $\mu$ 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 $15 .$

Accepted Answer

A)  3 
B)  $7.5$ 
C)  5 
D)  15 
E)  $3.8$ 
A)  3 
B)  $7.5$ 
C)  5 
D)  15 
E)  $3.8$

Find the centroid of the region bounded by the given curves. Select the correct answer. $y = 2 \sin 5 x , y = 0 , x = 0 , x = \frac { \pi } { 5 }$

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. $f ( x ) = x - 2 \sqrt { x } ; [ 0,3 ]$

Find the area of the surface obtained by rotating the circle $x ^ { 2 } + y ^ { 2 } = 7 ^ { 2 }$ about the line $y = 7$ .

Find the centroid of the region bounded by the curves. $y = 9 \ln 2 x , y = 0 , x = \frac { e } { 2 }$

Find the arc length of the graph of the given equation on the specified interval. $y = \frac { 2 } { 3 } \left( x ^ { 2 } + 1 \right) ^ { 3 / 2 } , [ 2,3 ]$

Find the center of mass of the lamina of the region shown if the density of the circular lamina is six times that of the rectangular lamina.

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

Write an integral giving the area of the surface obtained by revolving the curve about the $x$ -axis. (Do not evaluate the integral.) $y = \frac { 2 } { x } \text { on } [ 4,5 ]$

Find the centroid of the region bounded by the graphs of the given equations. $y = | x | \sqrt { 16 - x ^ { 2 } } , \quad y = 0 , \quad x = - 4 , \quad x = 4$

Find the centroid of the region bounded by the graphs of the given equations. $y = 15 - x ^ { 2 } , \quad y = 3 - x$

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 1,700 units.

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

Find the centroid of the region bounded by the given curves. $y = x ^ { 3 } , x + y = 2 , x = 0$

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

The standard deviation for a random variable with probability density function $f$ and mean $\mu$ 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 $15 .$

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Exam 8: Further Applications of Integration

Find the centroid of the region bounded by the given curves. Select the correct answer. y=2sin⁡5x,y=0,x=0,x=π5y = 2 \sin 5 x , y = 0 , x = 0 , x = \frac { \pi } { 5 }y=2sin5x,y=0,x=0,x=5π​

Find the area of the surface obtained by rotating the circle x2+y2=72x ^ { 2 } + y ^ { 2 } = 7 ^ { 2 }x2+y2=72 about the line y=7y = 7y=7 .

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 62.4lb/ft362.4 \mathrm { lb } / \mathrm { ft } ^ { 3 }62.4lb/ft3 .)

Find the centroid of the region bounded by the curves. y=9ln⁡2x,y=0,x=e2y = 9 \ln 2 x , y = 0 , x = \frac { e } { 2 }y=9ln2x,y=0,x=2e​

Find the arc length of the graph of the given equation on the specified interval. y=23(x2+1)3/2,[2,3]y = \frac { 2 } { 3 } \left( x ^ { 2 } + 1 \right) ^ { 3 / 2 } , [ 2,3 ]y=32​(x2+1)3/2,[2,3]

Find the center of mass of the lamina of the region shown if the density of the circular lamina is six times that of the rectangular lamina.

Find the area of the surface obtained by revolving the given curve about the yyy -axis. x=y3 on [0,2]x = y ^ { 3 } \text { on } [ 0,2 ]x=y3 on [0,2]

Write an integral giving the area of the surface obtained by revolving the curve about the xxx -axis. (Do not evaluate the integral.) y=2x on [4,5]y = \frac { 2 } { x } \text { on } [ 4,5 ]y=x2​ on [4,5]

A vertical plate is submerged in water (the surface of the water coincides with the xxx -axis). Find the force exerted by the water on the plate. (The weight density of water is 62.4lb/ft362.4 \mathrm { lb } / \mathrm { ft } ^ { 3 }62.4lb/ft3 .)

Find the centroid of the region bounded by the graphs of the given equations. y=∣x∣16−x2,y=0,x=−4,x=4y = | x | \sqrt { 16 - x ^ { 2 } } , \quad y = 0 , \quad x = - 4 , \quad x = 4y=∣x∣16−x2​,y=0,x=−4,x=4

Find the centroid of the region bounded by the graphs of the given equations. y=15−x2,y=3−xy = 15 - x ^ { 2 } , \quad y = 3 - xy=15−x2,y=3−x

The marginal revenue from producing xxx 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 1,700 units.

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 62.4lb/ft362.4 \mathrm { lb } / \mathrm { ft } ^ { 3 }62.4lb/ft3 .)

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

Find the centroid of the region bounded by the given curves. y=x3,x+y=2,x=0y = x ^ { 3 } , x + y = 2 , x = 0y=x3,x+y=2,x=0

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 0 \leq x \leq \frac { 9 \pi } { 2 }y=2exsinx,0≤x≤29π​

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

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

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Find the centroid of the region bounded by the given curves. Select the correct answer. $y = 2 \sin 5 x , y = 0 , x = 0 , x = \frac { \pi } { 5 }$

Find the area of the surface obtained by rotating the circle $x ^ { 2 } + y ^ { 2 } = 7 ^ { 2 }$ about the line $y = 7$ .

Find the centroid of the region bounded by the curves. $y = 9 \ln 2 x , y = 0 , x = \frac { e } { 2 }$

Find the arc length of the graph of the given equation on the specified interval. $y = \frac { 2 } { 3 } \left( x ^ { 2 } + 1 \right) ^ { 3 / 2 } , [ 2,3 ]$

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

Write an integral giving the area of the surface obtained by revolving the curve about the $x$ -axis. (Do not evaluate the integral.) $y = \frac { 2 } { x } \text { on } [ 4,5 ]$

Find the centroid of the region bounded by the graphs of the given equations. $y = | x | \sqrt { 16 - x ^ { 2 } } , \quad y = 0 , \quad x = - 4 , \quad x = 4$

Find the centroid of the region bounded by the graphs of the given equations. $y = 15 - x ^ { 2 } , \quad y = 3 - x$

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 1,700 units.

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

Find the centroid of the region bounded by the given curves. $y = x ^ { 3 } , x + y = 2 , x = 0$

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