Question 1

Solve the problem.
-A force of 4 N will stretch a rubber band 5 cm. Assuming Hooke's law applies, how much work is done on the rubber band by a 12 N force?

Accepted Answer

A)  9000 J 
B)  0.9 J 
C)  0.3 J 
D)  0.1 J 
A)  9000 J 
B)  0.9 J 
C)  0.3 J 
D)  0.1 J

Question 2

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis.
-$$y = 3 \csc x , y = 0 , x = \frac { \pi } { 4 } , x = \frac { 3 \pi } { 4 }$$

Accepted Answer

A)  $9 \pi$ 
B)  $27 \pi$ 
C)  $18 \pi$ 
D)  $6 \pi$ 
A)  $9 \pi$ 
B)  $27 \pi$ 
C)  $18 \pi$ 
D)  $6 \pi$

Question 3

Solve the problem.
-The base of a rectangular tank measures $7 \mathrm { ft }$ by $14 \mathrm { ft }$. The tank is $17 \mathrm { ft }$ tall, and its top is $10 \mathrm { ft }$ below ground level. The tank is full of water weighing $62.4 \mathrm { lb } / \mathrm { ft } ^ { 3 }$. How much work does it take to empty the tank by pumping the water to ground level? Give your answer to the nearest $\mathrm { ft }$ ' $\mathrm { lb }$.

Accepted Answer

A)  883,646 ft · lb 
B)  3,690,523 ft · lb 
C)  883,816 ft · lb 
D)  1,923,230 ft · lb 
A)  883,646 ft · lb 
B)  3,690,523 ft · lb 
C)  883,816 ft · lb 
D)  1,923,230 ft · lb

Question 4

Find the center of mass of a thin plate covering the given region with the given density function.
-The region enclosed by the parabolas $y = 50 - x ^ { 2 }$ and $y = x ^ { 2 }$, with density $\delta ( x ) = x ^ { 2 }$

Accepted Answer

A)  $\bar { x } = 0 , \bar { y } = 25$ 
B)  $\bar { x } = 25 , \bar { y } = 0$ 
C)  $\bar { x } = 0 , \bar { y } = 50$ 
D)  $\bar { x } = 0 , \bar { y } = 20$ 
A)  $\bar { x } = 0 , \bar { y } = 25$ 
B)  $\bar { x } = 25 , \bar { y } = 0$ 
C)  $\bar { x } = 0 , \bar { y } = 50$ 
D)  $\bar { x } = 0 , \bar { y } = 20$

Question 5

Find the area of the surface generated by revolving the curve about the indicated axis.
-$$y = \sqrt { 4 x - x ^ { 2 } } , 0.5 \leq x \leq 1.5 \text {; } x \text {-axis }$$

Accepted Answer

A)  $5 \pi$ 
B)  $3 \pi$ 
C)  $\pi$ 
D)  $4 \pi$ 
A)  $5 \pi$ 
B)  $3 \pi$ 
C)  $\pi$ 
D)  $4 \pi$

Question 6

Provide an appropriate response.
-The region bounded by the lines x = 2, x = 6, y = -2, and y = 1 is revolved about the y-axis to form a solid.
Explain how you could use elementary geometry formulas to verify the volume of this solid.

Accepted Answer

The resulting solid is a cylin

Question 7

Find the fluid force exerted against the vertically submerged flat surface depicted in the diagram. Assume arbitrary
units, and call the weight-density of the fluid w.
-

Accepted Answer

A)  $2058 \mathrm { w }$ 
B)  $\frac { 1029 } { 2 } w$ 
C)  $1029 w$ 
D)  $\frac { 343 } { 2 } \mathrm { w }$ 
A)  $2058 \mathrm { w }$ 
B)  $\frac { 1029 } { 2 } w$ 
C)  $1029 w$ 
D)  $\frac { 343 } { 2 } \mathrm { w }$

Question 8

Find the length of the curve.
-$$y = \int _ { 1 } ^ { x } \sqrt { t ^ { 2 } - 1 } d t , 3 \leq x \leq 6$$

Accepted Answer

A)  18 
B)  $\frac { 27 } { 2 }$ 
C)  9 
D)  27 
A)  18 
B)  $\frac { 27 } { 2 }$ 
C)  9 
D)  27

Question 9

Solve the problem.
-A rescue cable attached to a helicopter weighs 2 lb/ft. A 170-lb man grabs the end of the rope and is pulled from the ocean into the helicopter. How much work is done in lifting the man if the helicopter is 50 ft above the
Water?

Accepted Answer

A)  2670 ft · lb 
B)  13,500 ft · lb 
C)  8600 ft · lb 
D)  11,000 ft · lb 
A)  2670 ft · lb 
B)  13,500 ft · lb 
C)  8600 ft · lb 
D)  11,000 ft · lb

Question 10

Find the volume of the solid generated by revolving the shaded region about the given axis.
-About the $y$-axis

$x = \frac { y ^ { 2 } } { 3 }$

Accepted Answer

A)  $18 \pi$ 
B)  $\frac { 45 } { 2 } \pi$ 
C)  $\frac { 108 } { 5 } \pi$ 
D)  $\frac { 27 } { 5 } \pi$ 
A)  $18 \pi$ 
B)  $\frac { 45 } { 2 } \pi$ 
C)  $\frac { 108 } { 5 } \pi$ 
D)  $\frac { 27 } { 5 } \pi$

Question 11

Solve the problem.
-A rectangular swimming pool has a parabolic drain plate at the bottom of the pool. The drain plate is shaped like the region between $y = \frac { 1 } { 2 } x ^ { 2 }$ and the line $y = \frac { 1 } { 2 }$ from $x = - 1$ to $x = 1$. The pool is $10 \mathrm { ft }$ by $20 \mathrm { ft }$ and $8 \mathrm { ft }$ deep. If the drain plate is designed to withstand a fluid force of $200 \mathrm { lb }$, how high can the pool be filled without exceeding this limitation?

Accepted Answer

A)  5 ft 1 in. 
B)  5 ft 6 in. 
C)  5 ft 9 in. 
D)  4 ft 11 in. 
A)  5 ft 1 in. 
B)  5 ft 6 in. 
C)  5 ft 9 in. 
D)  4 ft 11 in.

Question 12

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis.
-$$y = \frac { 1 } { \sqrt { x } } , y = 0 , x = 1 , x = 6$$

Accepted Answer

A)  $\sqrt { 6 } \pi$ 
B)  $\frac { \pi } { 2 } ( \ln 6 )$ 
C)  $\frac { \pi } { 6 }$ 
D)  $\pi ( \ln 6 )$ 
A)  $\sqrt { 6 } \pi$ 
B)  $\frac { \pi } { 2 } ( \ln 6 )$ 
C)  $\frac { \pi } { 6 }$ 
D)  $\pi ( \ln 6 )$

Question 13

Solve the problem.
-A water tank is formed by revolving the curve $y = 3 x ^ { 4 }$ about the $y$-axis. Water drains from the tank through a small hole in the bottom of the tank. At what constant rate does the water level, $\mathrm { y }$, fall? (Use Torricelli's Law: $\mathrm { dV } / \mathrm { dt } = - \mathrm { m } \sqrt { \mathrm { y } }$.)

Accepted Answer

A)  $\frac { \mathrm { dy } } { \mathrm { dt } } = \frac { - \mathrm { m } \pi } { \sqrt { 3 } }$ 
B)  $\frac { \mathrm { dy } } { \mathrm { dt } } = \frac { - \mathrm { m } \sqrt { 3 } } { \pi }$ 
C)  $\frac { \mathrm { dy } } { \mathrm { dt } } = \frac { - \pi } { \mathrm { m } \sqrt { 3 } }$ 
D)  $\frac { d y } { d t } = \frac { - \sqrt { 3 } } { m \pi }$ 
A)  $\frac { \mathrm { dy } } { \mathrm { dt } } = \frac { - \mathrm { m } \pi } { \sqrt { 3 } }$ 
B)  $\frac { \mathrm { dy } } { \mathrm { dt } } = \frac { - \mathrm { m } \sqrt { 3 } } { \pi }$ 
C)  $\frac { \mathrm { dy } } { \mathrm { dt } } = \frac { - \pi } { \mathrm { m } \sqrt { 3 } }$ 
D)  $\frac { d y } { d t } = \frac { - \sqrt { 3 } } { m \pi }$

Question 14

Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the y-axis.
-y = 3x, y = 6x, x = 3

Accepted Answer

A)  $54 \pi$ 
B)  $18 \pi$ 
C)  $\frac { 243 } { 2 } \pi$ 
D)  $27 \pi$ 
A)  $54 \pi$ 
B)  $18 \pi$ 
C)  $\frac { 243 } { 2 } \pi$ 
D)  $27 \pi$

Question 15

Set up an integral for the length of the curve.
-$$y = x ^ { 4 } , 0 \leq x \leq 1$$

Accepted Answer

A)  $\int _ { 0 } ^ { 1 } \sqrt { 1 + 4 x ^ { 3 } } d x$ 
B)  $\int _ { 0 } ^ { 1 } \sqrt { 1 + 16 x ^ { 8 } } d x$ 
C)  $\int _ { 0 } ^ { 1 } \sqrt { 1 + 4 x ^ { 6 } } d x$ 
D)  $\int _ { 0 } ^ { 1 } \sqrt { 1 + 16 x ^ { 6 } } d x$ 
A)  $\int _ { 0 } ^ { 1 } \sqrt { 1 + 4 x ^ { 3 } } d x$ 
B)  $\int _ { 0 } ^ { 1 } \sqrt { 1 + 16 x ^ { 8 } } d x$ 
C)  $\int _ { 0 } ^ { 1 } \sqrt { 1 + 4 x ^ { 6 } } d x$ 
D)  $\int _ { 0 } ^ { 1 } \sqrt { 1 + 16 x ^ { 6 } } d x$

Question 16

Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves
about the given lines.
-$y = 25 - x ^ { 2 } , \quad y = 25 , \quad x = 5$; revolve about the line $y = 25$

Accepted Answer

A)  $\frac { 4375 } { 3 } \pi$ 
B)  $625 \pi$ 
C)  $\frac { 125 } { 3 } \pi$ 
D)  $\frac { 5000 } { 3 } \pi$ 
A)  $\frac { 4375 } { 3 } \pi$ 
B)  $625 \pi$ 
C)  $\frac { 125 } { 3 } \pi$ 
D)  $\frac { 5000 } { 3 } \pi$

Question 17

Solve the problem.
-It takes a force of 12,000 lb to compress a spring from its free height of 15 in. to its fully compressed height of 10 in. How much work does it take to compress the spring the first inch?

Accepted Answer

A)  600 in.-lb 
B)  1200 in.-lb 
C)  2400 in.-lb 
D)  120,000 in.-lb 
A)  600 in.-lb 
B)  1200 in.-lb 
C)  2400 in.-lb 
D)  120,000 in.-lb

Question 18

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis.
-y = 3x, y = 3, x = 0

Accepted Answer

A)  $6 \pi$ 
B)  $\frac { 3 } { 2 } \pi$ 
C)  $9 \pi$ 
D)  $3 \pi$ 
A)  $6 \pi$ 
B)  $\frac { 3 } { 2 } \pi$ 
C)  $9 \pi$ 
D)  $3 \pi$

Question 19

Find the volume of the described solid.
-The base of the solid is the disk $x ^ { 2 } + y ^ { 2 } \leq 9$. The cross sections by planes perpendicular to the $y$-axis between $y = - 3$ and $y = 3$ are isosceles right triangles with one leg in the disk.

Accepted Answer

A)  90 
B)  18 
C)  72 
D)  144 
A)  90 
B)  18 
C)  72 
D)  144

Question 20

Find the area of the surface generated by revolving the curve about the indicated axis.
-$$y = \frac { e ^ { x } + e ^ { - x } } { 2 } , 0 \leq x \leq \ln 4 ; x - a x i s$$

Accepted Answer

A)  $\frac { 15 } { 8 } \pi$ 
B)  $\pi \ln 4$ 
C)  $\pi \left( \frac { 255 } { 64 } + \ln 4 \right)$ 
D)  $\pi \left( \frac { 289 } { 32 } + \ln 4 \right)$ 
A)  $\frac { 15 } { 8 } \pi$ 
B)  $\pi \ln 4$ 
C)  $\pi \left( \frac { 255 } { 64 } + \ln 4 \right)$ 
D)  $\pi \left( \frac { 289 } { 32 } + \ln 4 \right)$

Solve the problem. -A force of 4 N will stretch a rubber band 5 cm. Assuming Hooke's law applies, how much work is done on the rubber band by a 12 N force?

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. - $y = 3 \csc x , y = 0 , x = \frac { \pi } { 4 } , x = \frac { 3 \pi } { 4 }$

Find the center of mass of a thin plate covering the given region with the given density function. -The region enclosed by the parabolas $y = 50 - x ^ { 2 }$ and $y = x ^ { 2 }$ , with density $\delta ( x ) = x ^ { 2 }$

Find the area of the surface generated by revolving the curve about the indicated axis. - $y = \sqrt { 4 x - x ^ { 2 } } , 0.5 \leq x \leq 1.5 \text {; } x \text {-axis }$

Provide an appropriate response. -The region bounded by the lines x = 2, x = 6, y = -2, and y = 1 is revolved about the y-axis to form a solid. Explain how you could use elementary geometry formulas to verify the volume of this solid.

Find the fluid force exerted against the vertically submerged flat surface depicted in the diagram. Assume arbitrary units, and call the weight-density of the fluid w. -

Find the length of the curve. - $y = \int _ { 1 } ^ { x } \sqrt { t ^ { 2 } - 1 } d t , 3 \leq x \leq 6$

Solve the problem. -A rescue cable attached to a helicopter weighs 2 lb/ft. A 170-lb man grabs the end of the rope and is pulled from the ocean into the helicopter. How much work is done in lifting the man if the helicopter is 50 ft above the Water?

Find the volume of the solid generated by revolving the shaded region about the given axis. -About the $y$ -axis $Find the volume of the solid generated by revolving the shaded region about the given axis. -About the y -axis x = \frac { y ^ { 2 } } { 3 }$ $x = \frac { y ^ { 2 } } { 3 }$

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. - $y = \frac { 1 } { \sqrt { x } } , y = 0 , x = 1 , x = 6$

Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the y-axis. -y = 3x, y = 6x, x = 3

Set up an integral for the length of the curve. - $y = x ^ { 4 } , 0 \leq x \leq 1$

Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves about the given lines. - $y = 25 - x ^ { 2 } , \quad y = 25 , \quad x = 5$ ; revolve about the line $y = 25$

Solve the problem. -It takes a force of 12,000 lb to compress a spring from its free height of 15 in. to its fully compressed height of 10 in. How much work does it take to compress the spring the first inch?

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. -y = 3x, y = 3, x = 0

Find the volume of the described solid. -The base of the solid is the disk $x ^ { 2 } + y ^ { 2 } \leq 9$ . The cross sections by planes perpendicular to the $y$ -axis between $y = - 3$ and $y = 3$ are isosceles right triangles with one leg in the disk.

Find the area of the surface generated by revolving the curve about the indicated axis. - $y = \frac { e ^ { x } + e ^ { - x } } { 2 } , 0 \leq x \leq \ln 4 ; x - a x i s$

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Exam 7: Applications of Definite Integrals