Deck 6: Inverse Functions

Question 1

If 10 J of work are needed to stretch a spring from 10 cm to 12 cm and another 20 J are needed to stretch it from 12 cm to 14 cm, what is the natural length of the spring? Round the answer to nearest integer.

A) $10 \text { cm }$
B) $11 \text { cm }$
C) $6 \mathrm{~cm}$
D) $8 \mathrm{~cm}$
E) $9 \mathrm{~cm}$

Answer 1

$9 \mathrm{~cm}$

Question 2

Find the average value of the function on the given interval.

Answer 2

Question 3

Find the volume of the solid obtained by revolving the region under the graph of on [1, ) about the x-axis.

Answer 3

not answered

Question 4

In a certain city the temperature $7\left(\mathrm{I}_{0} \mathrm{ul}\right)$ hours after 7 A.M. was modeled by the function $T(t)=45+30 \sin \frac{\pi t}{12}$ Find the average temperature to three decimal places during the period from 7 A.M. to 7 P.M.

A) $59.099^{\circ} \mathrm{F}$
B) $79.099^{\circ} \mathrm{F}$
C) $64.099^{\circ} \mathrm{F}$
D) $128.198 \mathrm{\circ} \mathrm{F}$
E) $74.099^{\circ} \mathrm{F}$

Question 5

A force of 30 N is required to maintain a spring stretched from its natural length of 12 cm to a length of 15 cm. How much work is done in stretching the spring from 11 cm to 24 cm?

A) $11.65 \mathrm{~J}$
B) $9.45 \mathrm{~J}$
C) $12.65 \mathrm{~J}$
D) $8.45 \mathrm{~J}$
E) $10.65 \mathrm{~J}$

Question 6

A bucket weighs 7 lb and a rope of negligible weight are used to draw water from a well that is 60 ft deep. The bucket starts with 50 lb of water and is pulled up at a rate of 10 ft/s, but water leaks out of a hole in the bucket at a rate of 0.5 lb/s. Find the work done in pulling the bucket to the top of the well.

A) $3140 \mathrm{ft}-\mathrm{lb}$
B) $2740 \mathrm{ft}-1 \mathrm{~b}$
C) $2640 \mathrm{ft}-1 \mathrm{~b}$
D) $4140 \mathrm{ft}-\mathrm{lb}$
E) $3640 \mathrm{ft}-\mathrm{lb}$

Question 7

Find the number(s) a such that the average value of the function on the interval is equal to .

Question 8

Find the average value of the function on the given interval. $h(x)=7 e^{\sin x} \cos x,\left[0, \frac{\pi}{2}\right]$

A) $14 e^{3}-1$
B) $16(e-1)$
C) $13 \pi\left(e^{3}-1\right)$
D) $\frac{15}{\pi}(e-1)$
E) $\frac{14}{\pi}(e-1)$

Question 9

Find the average value of the function on the interval .

Question 10

A tank is full of water. Find the work required to pump the water out of the outlet.

A) $W=405000 \mathrm{~J}$
B) $W=440000 \mathrm{~J}$
C) $W=4435000 \mathrm{~J}$
D) $W=4430000 \mathrm{~J}$
E) None of these

Question 11

Find the average value of the function $f(t)=9 t \sin \left(t^{2}\right)$ on the interval $[0,20]$ . Round your answer to 3 decimal places.

A) 9.342
B) 0.45
C) 0.432
D) $0.342$
E) 18

Question 12

A spring has a natural length of 20 cm. If a force of 25 N is required to keep it stretched to a length of 30 cm, how much work is required to stretch it from 20 cm to 32 cm?

A) $1.80 \mathrm{~J}$
B) $3.80 \mathrm{~J}$
C) $5.80 \mathrm{~J}$
D) $4.80 \mathrm{~J}$
E) $2.80 \mathrm{~J}$

Question 13

The Mean Value Theorem for Integrals says that if $f(t)$ is continuous on [ $a$ , $b$ ], then there exists a number m in [ $a$ , $b$ ] such that $f(m)=f_{a v e}=\frac{1}{a-b} \int_{a}^{b} f(t) d t .$

Question 14

The linear density of a $48$ m long rod is $\frac{72}{\sqrt{x+1}} \mathrm{~kg} / \mathrm{m}$ where x is measured in meters from one end of the rod. Find the average density of the rod.

A) $\rho_{\text {ave }}=21 \mathrm{~kg} / \mathrm{m}$
B) $\rho_{\text {ave }}=11 \mathrm{~kg} / \mathrm{m}$
C) $\rho_{\text {ave }}=41 \mathrm{~kg} / \mathrm{m}$
D) $\rho_{\text {ave }}=26 \mathrm{~kg} / \mathrm{m}$
E) $\rho_{\text {ave }}=23 \mathrm{~kg} / \mathrm{m}$

Question 15

A heavy rope, 40 ft long, weighs $0.8$ lb/ft and hangs over the edge of a building 110 ft high. How much work is done in pulling the rope to the top of the building?

A) $710 \mathrm{ft}-\mathrm{lb}$
B) $730 \mathrm{ft}-\mathrm{lb}$
C) $750 \mathrm{ft}-\mathrm{lb}$
D) $740 \mathrm{ft}-\mathrm{lb}$
E) $640 \mathrm{ft}-\mathrm{lb}$

Question 16

Find the average value of the function $f(t)=8 t \sin t^{2}$ on the interval $[0, \sqrt{\pi}]$ .

A) $\frac{16}{\pi}$
B) $\frac{8}{\pi}$
C) $\frac{8}{\sqrt{\pi}}$
D) $\frac{1}{\pi}$
E) $\frac{16}{\sqrt{\pi}}$

Question 17

The temperature of a metal rod, 6 m long, is 5 x (in degree Celsius) at a distance x meters from one end of the rod. What is the average temperature of the rod?

A) $T_{\text {ave }}=32^{\circ} \mathrm{C}$
B) $T_{\text {ave }}=56^{\circ} \mathrm{C}$
C) $T_{\text {ave }}=15^{\circ} \mathrm{C}$
D) $T_{\text {ave }}=62^{\circ} \mathrm{C}$
E) $T_{\text {ave }}=50^{\circ} \mathrm{C}$

Question 18

Find the number(s) a such that the average value of the function $f(x)=50-28 x+3 x^{2}$ on the interval $[0, a]$ is equal to 10.

A) $a=10$
B) $a=-4$
C) $a=-10$
D) $a=4$
E) $a=-10$ .

Question 19

Newton's Law of Gravitation states that two bodies with masses $m_{1} \text { and } m_{2}$ attract each other with a force $F=G \frac{m_{1} m_{2}}{r^{2}}$ where r is the distance between the bodies and G is the gravitation constant. If one of bodies is fixed, find the work needed to move the other from $r=m$ to $r=h$ .

A) $W=G m_{1} m_{2}\left(\frac{1}{m^{2}}-\frac{1}{h^{2}}\right)$
B) $W=G m_{1} m_{2}\left(\frac{1}{h^{2}}-\frac{1}{m^{2}}\right)$
C) $W=G m_{1} m_{2}\left(\frac{1}{h}-\frac{1}{m}\right)$
D) $W=G m_{1} m_{2}\left(\frac{1}{m^{2}}+\frac{1}{h^{2}}\right)$
E) $W=G m_{1} m_{2}\left(\frac{1}{m}-\frac{1}{h}\right)$

Question 20

The velocity v of blood that flows in a blood vessel with radius and length l at a distance from the central axis is where P is the pressure difference between the ends of the vessel and q is the viscosity of the blood. Find the average velocity (with respect to r) over the interval

Question 21

Use the method of cylindrical shells to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated axis. Sketch the region and a representative rectangle.
y = , y = 0, x = 2, x = 5; the y-axis

Question 22

Suppose you make napkin rings by drilling holes with different diameters through two wooden balls (which also have different diameters). You discover that both napkin rings have the same height h as shown in the figure. Use cylindrical shells to compute the volume of a napkin ring created by drilling a hole with radius d through the center of a sphere of radius D and express the answer in terms of $t$ .

A) $V=\frac{1}{3} \pi t^{3}$
B) $V=\frac{1}{4} \pi t^{3}$
C) $V=\frac{1}{3} \pi t^{2}$
D) $V=\frac{1}{6} \pi t^{2}$
E) $V=\frac{1}{6} \pi t^{3}$

Question 23

Use the method of cylindrical shells to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated axis. Sketch the region and a representative rectangle.
y = 3 , y = 0, x = 1; the y-axis

Question 24

Use cylindrical shells to find the volume of the solid. A sphere of radius $r$ .

A) $V=\frac{1}{3} \pi r^{3}$
B) $V=\frac{4}{3} \pi r^{3}$
C) $V=\frac{7}{3} \pi r^{3}$
D) $V=\frac{2}{3} \pi r^{3}$
E) $V=\frac{5}{3} \pi r^{3}$

Question 25

An aquarium 6 m long, 1 m wide, and 1 m deep is full of water. Find the work needed to pump half of the water out of the aquarium. (Use the facts that the density of water is $1000 \mathrm{~kg} / \mathrm{m}^{3} \text { and } g \approx 9.8 \text {. }$ )

A) $14700 \mathrm{~J}$
B) $7350 \mathrm{~J}$
C) $29400 \mathrm{~J}$
D) $58800 \mathrm{~J}$
E) $9800 \mathrm{~J}$

Question 26

If a force of 6 lbs is required to hold a spring stretched 5 inches beyond its natural length, then 38.4 lb-in. of work is done in stretching it from its natural length to 8 in. beyond its natural length.

Question 27

Use the Midpoint Rule with n = 4 to estimate the volume obtained by rotating about the region under the y-axis the region under the curve. $y=\tan x, \quad 0 \leq x \leq \frac{\pi}{4}$ Select the correct answer. The choices are rounded to the nearest hundredth.

A) $V=0.156$
B) $V=1.851$
C) $V=0.825$
D) $V=1.142$
E) $V=0.491$

Question 28

Use a graphing utility to (a) plot the graphs of the given functions, (b) find the approximate x-coordinates of the points of intersection of the graphs, and (c) find an approximation of the volume of the solid obtained by revolving the region bounded by the graphs of the functions about the y-axis. Round answers to two decimal places.
y = x, y = $x^{5}$ - $x^{2}$ , x $\ge$ 0

Question 29

A particle moves a distance of 150 ft along a straight line. As it moves, it is acted upon by a constant force of magnitude 15 lb in a direction opposite to that of the motion. What is the work done by the force?

A) $-\frac{1}{2250}$ ft-lb
B) $-10$ ft-lb
C) -2250 ft-lb
D) $-\frac{1}{10}$ ft-lb

Question 30

In a steam engine the pressure and volume of steam satisfy the equation , where k is a constant. (This is true for adiabatic expansion, that is, expansion in which there is no heat transfer between the cylinder and its surroundings.) Calculate the work done by the engine (in ft-lb) during a cycle when the steam starts at a pressure of and a volume of and expands to a volume of Use the fact that the work done by the gas when the volume expands from to volume is .

Question 31

Use the method of cylindrical shells to find the volume of solid obtained by rotating the region bounded by the given curves about the x-axis. $y=x^{2}, y=0, x=1, x=4 ; \text { about } x=1$

A) $\frac{255 \pi}{2}$
B) $3413 \pi$
C) $1020 \pi$
D) $\frac{2 \pi}{255}$
E) $255 \pi$

Question 32

Use the method of disks or washers, or the method of cylindrical shells to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated axis. Sketch the region and a representative rectangle. $y=2 \sqrt{x}$ , $y=x-3$ , $y=0$ , the x-axis

A) $45 \pi$
B) $135 \pi$
C) $180 \pi$
D) $90 \pi$

Question 33

Find the work done in pushing a car a distance of m while exerting a constant force of N.

Question 34

The base of a solid is a circular disk with radius . Find the volume of the solid if parallel cross-sections perpendicular to the base are isosceles right triangles with hypotenuse lying along the base.

Question 35

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated line. Sketch the region and a representative rectangle. y = 25 - $x^{2}$ , y = 0; the line x = - 5

A) $\frac{5000}{3}$ $\pi$
B) $1250$ $\pi$
C) $\frac{2500}{3}$ $\pi$
D) $625$ $\pi$

Question 36

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis.

Question 37

Use the method of disks or washers, or the method of cylindrical shells to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated axis. Sketch the region and a representative rectangle.
y = 2 , y = 4x - 2, y = 8; the y-axis

Question 38

Use the method of cylindrical shells to find the volume of solid obtained by rotating the region bounded by the given curves about the x-axis.

Question 39

Sketch a plane region and indicate the axis about which it is revolved so that the resulting solid of revolution (found using the shell method) is given by the integral. 2 $\pi$ $\int_{0}^{1} y\left(y^{1 / 9}-y\right) d y$

A) x-axis
B) y-axis
C) x-axis
D) y-axis

Question 40

Use the method of cylindrical shells to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated axis. Sketch the region and a representative rectangle. y = $\sqrt{16-x^{2}}$ , y = -x + 4; the y-axis

A) $\frac{64}{3}$ $\pi$

B) 128 $\pi$

C) 128 $\pi$

D) $\frac{64}{3}$ $\pi$

Question 41

The height of a monument is $20$ m. A horizontal cross-section at a distance x meters from the top is an equilateral triangle with side $\frac{x}{4}$ meters. Find the volume of the monument.

A) $\frac{125 \sqrt{3}}{3} \mathrm{~m}^{3}$
B) $\frac{125 \sqrt{2}}{3} \mathrm{~m}^{3}$
C) $\frac{\sqrt{3}}{3} \mathrm{~m}^{3}$
D) $\frac{125 \sqrt{2}}{2} \mathrm{~m}^{3}$
E) $125 \sqrt{3} \mathrm{~m}^{3}$

Question 42

Sketch a graph to estimate the x-coordinates of the points of intersection of the given curves. Then use this information to estimate the volume of the solid obtained by rotating about the y axis the region enclosed by these curves. Rounded to the nearest hundredth. $y=0, y=-x^{4}+6 x^{3}-x^{2}+6 x$

A) $V=3,745.96 \pi$
B) $V=3,323.88 \pi$
C) $V=3,331.63 \pi$
D) $V=3,346.96 \pi$
E) $V=3,326.40 \pi$

Question 43

The base of S is a circular region with boundary curve Cross-sections perpendicular to the x axis are isosceles right triangles with hypotenuse in the base.
Find the volume of S.

Question 44

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. $y=5\left(x^{2}+4\right), y=5\left(12-x^{2}\right) ; \text { about } y=-5$

A) $9550 \pi$
B) $9525 \pi$
C) $9575 \pi$
D) $9600 \pi$
E) None of these

Question 45

Find the volume of the solid obtained by rotating the region bounded by $y=x^{3} \text { and } x=y^{3}$ about the x-axis.

A) $\frac{16}{7} \pi$
B) $\frac{16}{35} \pi$
C) $\frac{7}{2}$
D) $\frac{18}{35}$
E) $16 \pi$

Question 46

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated line. y = 4 - $x^{2}$ , y = 0; the line y = 5

A) $\frac{1088}{5}$ $\pi$
B) $\frac{1088}{15}$ $\pi$
C) $\frac{544}{15}$ $\pi$
D) $\frac{2176}{15}$ $\pi$

Question 47

Find the volume of the solid obtained by rotating the region bounded by $y=2 \sqrt[4]{x} \text { and } y=2 x$ about the line $y=2 .$

A) $\frac{16 \pi}{15}$
B) $\frac{2}{9}$
C) $\frac{1}{16}$
D) $\frac{\pi}{9}$
E) $\frac{8 \pi}{9}$

Question 48

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated line. Sketch the region and a representative rectangle.
y = , y = x - 1; the line x = 2

Question 49

Use a computer algebra system to find the exact volume of the solid obtained by rotating the region bounded by the given curves about the specified line.

Question 50

The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method. $x^{2}+(y-1)^{2}=1^{2}$ about the y-axis

A) $V=\frac{4}{3} \pi$
B) $V=\frac{8}{3} \pi$
C) $V=\frac{28}{3} \pi$
D) $V=\frac{1}{3} \pi$
E) $V=\frac{20}{3} \pi$

Question 51

Find the volume common to two spheres, each with radius r = if the center of each sphere lies on the surface of the other sphere.

Question 52

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the x-axis. y = cos x + 1, x = 0, y = 0, x = $\frac{1}{2}$ $\pi$

A) $\frac{3}{4}$ $\pi^{2}$ + 2 $\pi$
B) $\frac{3}{4}$ $\pi^{2}$ + 4 $\pi$
C)
$\pi^{2}$ + 4 $\pi$
D)
$\pi^{2}$ +2 $\pi$

Question 53

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations and inequalities about the y-axis. $x^{2}$ - $y^{2}$ = 16, x $\ge$ 0, y = - 4, y = 4

A) $256$ $\pi$
B) $\frac{512}{3}$ $\pi$
C) $128$ $\pi$
D) $\frac{256}{3}$ $\pi$

Question 54

Find the volume of the solid that is obtained by revolving the region about the line y = $\frac{5}{2}$ .

A) $\frac{89}{42}$ $\pi$

B) $\frac{89}{14}$ $\pi$

C) $\frac{89}{84}$ $\pi$

D) 89 $\pi$

Question 55

Use the method of disks or washers, or the method of cylindrical shells to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated axis. Sketch the region and a representative rectangle.
y = , y = 2x - 1, y = 4; the y-axis

Question 56

The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method. Round your answer to 3 decimal places. $y=x^{2}+3 x-10$ about the y-axis and $y=0$ .

A) 1763.213
B) None of these
C) 1752.016
D) 1760.025
E) 880.012

Question 57

Find the volume of a pyramid with height 4 and base an equilateral triangle with side a = 4 .

A) $\frac{16 \sqrt{3}}{3}$
B) $\frac{3 \sqrt{3}}{2}$
C) $16 \sqrt{3}$
D) $\frac{16 \sqrt{3}}{5}$
E) $\frac{\sqrt{3}}{10}$

Question 58

Find the volume of the solid that is obtained by revolving the region about the x-axis.

A) $\frac{199}{5}$ $\pi$

B) $\frac{597}{20}$ $\pi$

C) $\frac{199}{20}$ $\pi$

D) $\frac{199}{10}$ $\pi$

Question 59

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. $y=\ln x, y=1, y=3, x=0 \text {; about the } y \text { - axis }$

A) $\frac{\pi}{2}\left(e^{3}+1\right)$
B) $\frac{\pi}{2}\left(e^{6}-e^{2}\right)$
C) $\frac{e^{6}}{2}$
D) $\frac{e^{6}-e^{3}}{2}$
E) None

Question 60

Find the volume of the frustum of a pyramid with square base of side square top of side and height

Question 61

The base of S is the parabolic region Cross-sections perpendicular to the y axis are squares.
Find the volume of S.

Question 62

Find the volume of the solid obtained by rotating the region bounded by and about the y-axis.

Question 63

Sketch a plane region, and indicate the axis about which it is revolved so that the resulting solid of revolution has the volume given by the integral.

Question 64

Find the area of the shaded region.

A) $\frac{112}{3}$
B) $\frac{56}{3}$
C) $\frac{112}{9}$
D) $\frac{224}{3}$

Question 65

Graph the region between the curves and use your calculator to compute the area correct to five decimal places. $y=5 e^{1-x^{2}}, y=5 x^{4}$

A) 12.08127
B) 91.504
C) 3.141257
D) 3.66016
E) 18.3008

Question 66

Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis.

Question 67

Find the area of the region bounded by the given curves. $y=\cos x, y=\sin 3 x, x=0, x=\frac{\pi}{2}$

A) $\frac{1}{4}$
B) $\frac{1}{2}$
C) 4
D) 2
E) $\frac{1}{3}$

Question 68

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated axis. , , , ; the x-axis

Question 69

Find the volume of the solid obtained by rotating about the x-axis the region under the curve from x = to x = .

Question 70

Find the volume of the solid obtained by rotating the region bounded by and about the line

Question 71

Find the area of the shaded region.

A) 7
B) $\frac{7}{12}$
C) $\frac{1}{6}$
D) $\frac{7}{6}$

Question 72

Find the number b such that the line $y=b$ divides the region bounded by the curves $y=x^{2}$ and $y=5$ into two regions with equal area.

A) $5^{1 / 2}$
B) $5^{2 / 3}$
C) $\frac{2}{3}$
D) $5^{1 / 3}$
E) $\frac{1}{3}$

Question 73

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis.

Question 74

Find the volume of a cap of a sphere with radius r = and height h = 3 .

Question 75

Find the area of the region bounded by the given curves. $y=\sin \left(\frac{\pi x}{6}\right), y=x^{2}-6 x$

A) $16+\frac{12}{\pi}$
B) None of these
C) $\frac{-54}{3}-\frac{12}{\pi}$
D) $-54+\frac{\pi}{12}$
E) $\frac{-54}{3}+\frac{12}{\pi}$

Question 76

Use the method of cylindrical shells to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated axis. Sketch the region and a representative rectangle. , y = 0, x = 1, x = 12; the y-axis

Question 77

The volume of a solid torus (the donut-shaped solid shown in the figure) with r = 5 and R = 15 is $750 \pi^{2}$

Question 78

Use a graphing utility to (a) plot the graphs of the given functions and (b) find the approximate x-coordinates of the points of intersection of the graphs. Then find an approximation of the volume of the solid obtained by revolving the region bounded by the graphs of the functions about the x-axis. Round answers to two decimal places.
y = , y = 3 -

Question 79

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated axis.
y = , y = 0, x = 3, x = 5; the x-axis

Question 80

Find the area of the region bounded by the given curves. $y=x^{2}-6 x, y=5 x+12$

A) $2197$
B) $\frac{25}{3}$
C) $\frac{2197}{3}$
D) $\frac{2197}{6}$
E) 6