Question 1

Find the area of the surface obtained by revolving the given curve about the $x$-axis.
$$y = \frac { e ^ { x } + e ^ { - x } } { 2 } \text { on } [ 0 , \ln 4 ]$$

Accepted Answer

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

Question 2

Set up, but do not evaluate, an integral for the length of the curve.
$y = 6 e ^ { x } \sin x , \quad 0 \leq x \leq \frac { 9 \pi } { 2 }$

Accepted Answer

A)  $L = \int _ { 0 } ^ { 9 x / 2 } \sqrt { 1 - 36 e ^ { 2 x } ( 1 + \sin 2 x ) } d x$ 
B)  $L = \int _ { 0 } ^ { 9 x / 2 } \sqrt { 1 + 36 e ^ { 2 x } ( 1 + \sin 2 x ) } d x$ 
C) 
$L = \int _ { 0 } ^ { 9 x / 2 } \sqrt { 1 - 36 e ^ { 2 x } ( 1 - \sin 2 x ) } d x$ 
D) 
$L = \int _ { 0 } ^ { 9 x / 2 } \sqrt { 1 + 36 e ^ { 2 x } ( 1 - \sin 2 x ) } d x$ 
E) 
$$L = \int _ { 0 } ^ { 9 x / 2 } \sqrt { 1 + 36 e ^ { 2 x } ( 1 - \sin x ) } d x$$ 
A)  $L = \int _ { 0 } ^ { 9 x / 2 } \sqrt { 1 - 36 e ^ { 2 x } ( 1 + \sin 2 x ) } d x$ 
B)  $L = \int _ { 0 } ^ { 9 x / 2 } \sqrt { 1 + 36 e ^ { 2 x } ( 1 + \sin 2 x ) } d x$ 
C) 
$L = \int _ { 0 } ^ { 9 x / 2 } \sqrt { 1 - 36 e ^ { 2 x } ( 1 - \sin 2 x ) } d x$ 
D) 
$L = \int _ { 0 } ^ { 9 x / 2 } \sqrt { 1 + 36 e ^ { 2 x } ( 1 - \sin 2 x ) } d x$ 
E) 
$$L = \int _ { 0 } ^ { 9 x / 2 } \sqrt { 1 + 36 e ^ { 2 x } ( 1 - \sin x ) } d x$$

Question 3

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 4

Find the centroid of the region bounded by the graphs of the given equations.
$$y = 10 - x ^ { 2 } , \quad y = 2 - 2 x$$

Accepted Answer

A)  $( 0,1 )$ 
B)  $\left( \frac { 18 } { 5 } , 1 \right)$ 
C)  $\left( 1 , \frac { 18 } { 5 } \right)$ 
D)  $( 1,0 )$ 
A)  $( 0,1 )$ 
B)  $\left( \frac { 18 } { 5 } , 1 \right)$ 
C)  $\left( 1 , \frac { 18 } { 5 } \right)$ 
D)  $( 1,0 )$

Question 5

Find the area of the surface obtained by rotating the curve about the $y$-axis.
$$y = \frac { 1 } { 4 } x ^ { 2 } - \frac { 1 } { 2 } \ln x , 1 \leq x \leq 8$$

Accepted Answer

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

Question 6

Write an integral giving the arc length of the graph of the equation from $\mathbf { P }$ to $\mathbf { Q }$.
$$y = \frac { 1 } { x ^ { 2 } + 2 } ; \mathrm { P } \left( 1 , \frac { 1 } { 3 } \right) , \mathrm { Q } \left( 3 , \frac { 1 } { 11 } \right)$$

Accepted Answer

The answer of Write an integral giving the arc length...

Question 7

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 $10 .$

Accepted Answer

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

Question 8

The demand function for a commodity is given by
$$p = 2,000 - 0.1 x - 0.01 x ^ { 2 } .$$
Find the consumer surplus when the sales level is 75 .

Accepted Answer

The answer of The demand function for a commodity is...

Question 9

Use the arc length formula to find the length of the curve.
$$y = 5 - 3 x , - 7 \leq x \leq - 3$$

Accepted Answer

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

Question 10

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

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 11

Find the centroid of the region bounded by the graphs of the given equations.
$$y = x ^ { 2 / 3 } , \quad y = 0 , \quad x = 8$$

Accepted Answer

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

Question 12

$$\text { Find the centroid of the region shown in the figure. }$$

Accepted Answer

The answer of \[\text { Find the centroid of the...

Question 13

Find the length of the curve. Select the correct answer.
$$x = \frac { y ^ { 4 } } { 8 } + \frac { 1 } { 4 y ^ { 2 } } , 1 \leq y \leq 3$$

Accepted Answer

A)  $10.2222$ 
B)  $25.055$ 
C)  $13.05$ 
D)  $36.05$ 
E)  None of these 
A)  $10.2222$ 
B)  $25.055$ 
C)  $13.05$ 
D)  $36.05$ 
E)  None of these

Question 14

Find the centroid of the region bounded by the given curves. Select the correct answer.
$$y = 8 \sin 5 x , y = 0 , x = 0 , x = \frac { \pi } { 5 }$$

Accepted Answer

A)  $( - 0.3146,0.7854 )$ 
B)  $( 0.3146,25.1328 )$ 
C)  $( 2.5130,0.3146 )$ 
D)  $( - 2.5130,25.1328 )$ 
E)  $( 0.2513,2.5133 )$ 
A)  $( - 0.3146,0.7854 )$ 
B)  $( 0.3146,25.1328 )$ 
C)  $( 2.5130,0.3146 )$ 
D)  $( - 2.5130,25.1328 )$ 
E)  $( 0.2513,2.5133 )$

Question 15

Use differentials to approximate the arc length of the graph of the equation from $\mathbf { P }$ to $\mathbf { Q }$. Round answer to four decimal places.
$$y = x ^ { 3 } + 3 ; \mathrm { P } ( 1,4 ) , \quad \mathrm { Q } ( 1.2,4.728 )$$

Accepted Answer

The answer of Use differentials to approximate the arc length...

Question 16

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

Question 17

A trough is filled with a liquid of density $855 \mathrm {~kg} / \mathrm { m } ^ { 3 }$. The ends of the trough are equilateral triangles with sides $7 \mathrm {~m}$ long and vertex at the bottom. Find the hydrostatic force on one end of the trough.

Accepted Answer

A)  $2.5 \times 10 ^ { 5 } \mathrm {~N}$ 
B)  $5.5 \times 10 ^ { 5 } \mathrm {~N}$ 
C)  $4.5 \times 10 ^ { 5 } \mathrm {~N}$ 
D)  $6.5 \times 10 ^ { 5 } \mathrm {~N}$ 
E)  $3.5 \times 10 ^ { 5 } \mathrm {~N}$ 
A)  $2.5 \times 10 ^ { 5 } \mathrm {~N}$ 
B)  $5.5 \times 10 ^ { 5 } \mathrm {~N}$ 
C)  $4.5 \times 10 ^ { 5 } \mathrm {~N}$ 
D)  $6.5 \times 10 ^ { 5 } \mathrm {~N}$ 
E)  $3.5 \times 10 ^ { 5 } \mathrm {~N}$

Question 18

The marginal cost function $C ^ { \prime } ( x )$ is defined to be the derivative of the cost function. If the marginal cost of manufacturing $x$ units of a product is
$$C ^ { \prime } ( x ) = 0.009 x ^ { 2 } - 1.8 x + 9$$
(measured in dollars per unit) and the fixed start-up cost is $C ( 0 ) = 2,200,000$, use the Total Change Theorem to find the cost of producing the first 5,000 units.

Accepted Answer

A)  $\$ 354,445,000$ 
B)  $\$ 354,345,000$ 
C)  $\$ 354,845,000$ 
D)  $\$ 354,545,000$ 
E)  $\$ 354,745,000$ 
A)  $\$ 354,445,000$ 
B)  $\$ 354,345,000$ 
C)  $\$ 354,845,000$ 
D)  $\$ 354,545,000$ 
E)  $\$ 354,745,000$

Question 19

For a given commodity and pure competition, the number of units produced and the price per unit are determined as the coordinates of the point of intersection of the supply and demand curves. Given the demand curve
$$p = 60 - \frac { x } { 20 }$$
and the supply curve
$$p = 30 + \frac { x } { 30 } ,$$
find the consumer surplus.

Accepted Answer

The answer of For a given commodity and pure competition,...

Question 20

Use the Theorem of Pappus to find the volume of the solid obtained by revolving the region bounded by the graphs of $y = 16 - x ^ { 2 } , y = 16$, and $x = 4$ about the $y$-axis.

Accepted Answer

A)  $128 \pi$ 
B)  $\frac { 512 } { 3 } \pi$ 
C)  $\frac { 128 } { 3 } \pi$ 
D)  $32 \pi$ 
A)  $128 \pi$ 
B)  $\frac { 512 } { 3 } \pi$ 
C)  $\frac { 128 } { 3 } \pi$ 
D)  $32 \pi$

Find the area of the surface obtained by revolving the given curve about the $x$ -axis. $y = \frac { e ^ { x } + e ^ { - x } } { 2 } \text { on } [ 0 , \ln 4 ]$

Set up, but do not evaluate, an integral for the length of the curve. $y = 6 e ^ { x } \sin x , \quad 0 \leq x \leq \frac { 9 \pi } { 2 }$

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

Find the centroid of the region bounded by the graphs of the given equations. $y = 10 - x ^ { 2 } , \quad y = 2 - 2 x$

Find the area of the surface obtained by rotating the curve about the $y$ -axis. $y = \frac { 1 } { 4 } x ^ { 2 } - \frac { 1 } { 2 } \ln x , 1 \leq x \leq 8$

Write an integral giving the arc length of the graph of the equation from $\mathbf { P }$ to $\mathbf { Q }$ . $y = \frac { 1 } { x ^ { 2 } + 2 } ; \mathrm { P } \left( 1 , \frac { 1 } { 3 } \right) , \mathrm { Q } \left( 3 , \frac { 1 } { 11 } \right)$

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 $10 .$

The demand function for a commodity is given by $p = 2,000 - 0.1 x - 0.01 x ^ { 2 } .$ Find the consumer surplus when the sales level is 75 .

Use the arc length formula to find the length of the curve. $y = 5 - 3 x , - 7 \leq x \leq - 3$

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

Find the centroid of the region bounded by the graphs of the given equations. $y = x ^ { 2 / 3 } , \quad y = 0 , \quad x = 8$

$\text { Find the centroid of the region shown in the figure. }$ $\text { Find the centroid of the region shown in the figure. }$

Find the length of the curve. Select the correct answer. $x = \frac { y ^ { 4 } } { 8 } + \frac { 1 } { 4 y ^ { 2 } } , 1 \leq y \leq 3$

Find the centroid of the region bounded by the given curves. Select the correct answer. $y = 8 \sin 5 x , y = 0 , x = 0 , x = \frac { \pi } { 5 }$

Use differentials to approximate the arc length of the graph of the equation from $\mathbf { P }$ to $\mathbf { Q }$ . Round answer to four decimal places. $y = x ^ { 3 } + 3 ; \mathrm { P } ( 1,4 ) , \quad \mathrm { Q } ( 1.2,4.728 )$

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

A trough is filled with a liquid of density $855 \mathrm {~kg} / \mathrm { m } ^ { 3 }$ . The ends of the trough are equilateral triangles with sides $7 \mathrm {~m}$ long and vertex at the bottom. Find the hydrostatic force on one end of the trough.

Use the Theorem of Pappus to find the volume of the solid obtained by revolving the region bounded by the graphs of $y = 16 - x ^ { 2 } , y = 16$ , and $x = 4$ about the $y$ -axis.

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Exam 8: Further Applications of Integration

Find the area of the surface obtained by revolving the given curve about the xxx -axis. y=ex+e−x2 on [0,ln⁡4]y = \frac { e ^ { x } + e ^ { - x } } { 2 } \text { on } [ 0 , \ln 4 ]y=2ex+e−x​ on [0,ln4]

Set up, but do not evaluate, an integral for the length of the curve. y=6exsin⁡x,0≤x≤9π2y = 6 e ^ { x } \sin x , \quad 0 \leq x \leq \frac { 9 \pi } { 2 }y=6exsinx,0≤x≤29π​

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

Find the centroid of the region bounded by the graphs of the given equations. y=10−x2,y=2−2xy = 10 - x ^ { 2 } , \quad y = 2 - 2 xy=10−x2,y=2−2x

Find the area of the surface obtained by rotating the curve about the yyy -axis. y=14x2−12ln⁡x,1≤x≤8y = \frac { 1 } { 4 } x ^ { 2 } - \frac { 1 } { 2 } \ln x , 1 \leq x \leq 8y=41​x2−21​lnx,1≤x≤8

The demand function for a commodity is given by p=2,000−0.1x−0.01x2.p = 2,000 - 0.1 x - 0.01 x ^ { 2 } .p=2,000−0.1x−0.01x2. Find the consumer surplus when the sales level is 75 .

Use the arc length formula to find the length of the curve. y=5−3x,−7≤x≤−3y = 5 - 3 x , - 7 \leq x \leq - 3y=5−3x,−7≤x≤−3

Find the centroid of the region bounded by the graphs of the given equations. y=x2/3,y=0,x=8y = x ^ { 2 / 3 } , \quad y = 0 , \quad x = 8y=x2/3,y=0,x=8

Find the centroid of the region shown in the figure. \text { Find the centroid of the region shown in the figure. } Find the centroid of the region shown in the figure.

Find the length of the curve. Select the correct answer. x=y48+14y2,1≤y≤3x = \frac { y ^ { 4 } } { 8 } + \frac { 1 } { 4 y ^ { 2 } } , 1 \leq y \leq 3x=8y4​+4y21​,1≤y≤3

Find the centroid of the region bounded by the given curves. Select the correct answer. y=8sin⁡5x,y=0,x=0,x=π5y = 8 \sin 5 x , y = 0 , x = 0 , x = \frac { \pi } { 5 }y=8sin5x,y=0,x=0,x=5π​

Use differentials to approximate the arc length of the graph of the equation from P\mathbf { P }P to Q\mathbf { Q }Q . Round answer to four decimal places. y=x3+3;P(1,4),Q(1.2,4.728)y = x ^ { 3 } + 3 ; \mathrm { P } ( 1,4 ) , \quad \mathrm { Q } ( 1.2,4.728 )y=x3+3;P(1,4),Q(1.2,4.728)

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

A trough is filled with a liquid of density 855 kg/m3855 \mathrm {~kg} / \mathrm { m } ^ { 3 }855 kg/m3 . The ends of the trough are equilateral triangles with sides 7 m7 \mathrm {~m}7 m long and vertex at the bottom. Find the hydrostatic force on one end of the trough.

Use the Theorem of Pappus to find the volume of the solid obtained by revolving the region bounded by the graphs of y=16−x2,y=16y = 16 - x ^ { 2 } , y = 16y=16−x2,y=16 , and x=4x = 4x=4 about the yyy -axis.