Question 1

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

Accepted Answer

A)  $16+\frac{12}{\pi}$ 
B)  None of these 
C)  $\frac{-54}{3}-\frac{12}{\pi}$ 
D)  $-54+\frac{\pi}{12}$ 
E)  $\frac{-54}{3}+\frac{12}{\pi}$ 
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 2

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

Accepted Answer

A) True 
 B)False

Question 3

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 $r=h$ .

Accepted Answer

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)  $W=G m_{1} m_{2}\left(\frac{1}{m}-\frac{1}{h}\right)$ 
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 4

Find the volume of the frustum of a pyramid with square base of side $b=17$ square top of side $a=5,$ and height $h=19 .$

Accepted Answer

The answer of Find the volume of the frustum of...

Question 5

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.

Accepted Answer

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)  $3640 \mathrm{ft}-\mathrm{lb}$ 
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 6

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 $f(m)=f_{a v e}=\frac{1}{a-b} \int_{a}^{b} f(t) d t .$

Accepted Answer

A) True 
 B)False

Question 7

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated axis. $y=\frac{e^{x / 2}}{\left(1+e^{x}\right)^{5 / 2}}$ , $y=0$ , $x=0$ , $x=4$ ; the x-axis

Accepted Answer

The answer of Find the volume of the solid generated...

Question 8

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

Accepted Answer

The answer of Use the method of cylindrical shells to...

Question 9

In a steam engine the pressure and volume of steam satisfy the equation $P V^{1.4}=k$ , 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 $160 \mathrm{~lb} / \mathrm{in}^{2}$ and a volume of $100 \mathrm{~in}^{3}$ and expands to a volume of $800 \mathrm{~in}^{3} .$ Use the fact that the work done by the gas when the volume expands from $V_{1}$ to volume $V_{2}$ is $W=\int_{V_{1}}^{V_{2}} P d V$ .

Accepted Answer

The answer of In a steam engine the pressure and...

Question 10

Sketch the region bounded by the graphs of the given equations and find the area of that region.
y = $-x^{2}$ + 4, y = 2x + 1

Accepted Answer

The answer of Sketch the region bounded by the graphs...

Question 11

Find the average value of the function on the given interval. $g(x)=5 x e^{-x^{2}},[0,5]$

Accepted Answer

The answer of Find the average value of the function...

Question 12

Sketch the region bounded by the graphs of the given equations and find the area of that region. y = sec2 x + 6, y = 8, x = - $\frac{\pi}{4}$ , x = $\frac{\pi}{4}$

Accepted Answer

A)    $\pi$ - 2 
B)    $\pi$+8 
C)    $\frac{\pi}{2}$ - 1 
D)    2-$\pi$ 
A)    $\pi$ - 2 
B)    $\pi$+8 
C)    $\frac{\pi}{2}$ - 1 
D)    2-$\pi$

Question 13

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

Accepted Answer

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

Question 14

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

Accepted Answer

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

Question 15

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?

Accepted Answer

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)  $T_{\text {ave }}=50^{\circ} \mathrm{C}$ 
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 16

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 $t$ .

Accepted Answer

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)  $V=\frac{1}{6} \pi t^{3}$ 
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 17

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

Accepted Answer

A)  $W=405000 \mathrm{~J}$ 
B)  $W=440000 \mathrm{~J}$ 
C)  $W=4435000 \mathrm{~J}$ 
D)  $W=4430000 \mathrm{~J}$ 
E)  None of these 
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 18

Find the volume of the solid obtained by rotating the region bounded by $x=y^{2}$ and $x=5 y$ about the y-axis.

Accepted Answer

The answer of Find the volume of the solid obtained...

Question 19

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. $y=\sin x, y=0, x=4 \pi, x=11 \pi \text {; about the } y \text { - axis }$

Accepted Answer

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

Question 20

Sketch the region enclosed by $y=1+\sqrt{x} \text { and } y=\frac{4+x}{4}$ . Find t he area of the region.

Accepted Answer

A)  $\frac{32}{3}$ 
B)  $\frac{4 \sqrt{2}}{3}-4$ 
C)  $256 \sqrt{3}$ 
D)  $\frac{32}{3}$ $\sqrt{2}$ 
E)  $4 \sqrt{2}$ 
A)  $\frac{32}{3}$ 
B)  $\frac{4 \sqrt{2}}{3}-4$ 
C)  $256 \sqrt{3}$ 
D)  $\frac{32}{3}$ $\sqrt{2}$ 
E)  $4 \sqrt{2}$

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

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

Find the volume of the frustum of a pyramid with square base of side $b=17$ square top of side $a=5,$ and height $h=19 .$

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 $f(m)=f_{a v e}=\frac{1}{a-b} \int_{a}^{b} f(t) d t .$

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated axis. $y=\frac{e^{x / 2}}{\left(1+e^{x}\right)^{5 / 2}}$ , $y=0$ , $x=0$ , $x=4$ ; the x-axis

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

Sketch the region bounded by the graphs of the given equations and find the area of that region. y = $-x^{2}$ + 4, y = 2x + 1

Find the average value of the function on the given interval. $g(x)=5 x e^{-x^{2}},[0,5]$

Sketch the region bounded by the graphs of the given equations and find the area of that region. y = sec² x + 6, y = 8, x = - $\frac{\pi}{4}$ , x = $\frac{\pi}{4}$

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

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

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

Find the volume of the solid obtained by rotating the region bounded by $x=y^{2}$ and $x=5 y$ about the y-axis.

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. $y=\sin x, y=0, x=4 \pi, x=11 \pi \text {; about the } y \text { - axis }$

Sketch the region enclosed by $y=1+\sqrt{x} \text { and } y=\frac{4+x}{4}$ . Find t he area of the region.

Functions and Limits

Derivatives

Applications of Differentiation

Integrals

Applications of Integration

Techniques of Integration

Further Applications 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 6: Inverse Functions

Find the area of the region bounded by the given curves. y=sin⁡(πx6),y=x2−6xy=\sin \left(\frac{\pi x}{6}\right), y=x^{2}-6 xy=sin(6πx​),y=x2−6x

The volume of a solid torus (the donut-shaped solid shown in the figure) with r = 5 and R = 15 is 750π2750 \pi^{2}750π2

Find the volume of the frustum of a pyramid with square base of side b=17b=17b=17 square top of side a=5,a=5,a=5, and height h=19.h=19 .h=19.

The Mean Value Theorem for Integrals says that if f(t)f(t)f(t) is continuous on [ aaa , bbb ], then there exists a number m in [ aaa , bbb ] such that f(m)=fave=1a−b∫abf(t)dt.f(m)=f_{a v e}=\frac{1}{a-b} \int_{a}^{b} f(t) d t .f(m)=fave​=a−b1​∫ab​f(t)dt.

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated axis. y=ex/2(1+ex)5/2y=\frac{e^{x / 2}}{\left(1+e^{x}\right)^{5 / 2}}y=(1+ex)5/2ex/2​ , y=0y=0y=0 , x=0x=0x=0 , x=4x=4x=4 ; the x-axis

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis. y=x4,y=0,y=1; about the line x=4y=x^{4}, y=0, y=1 ; \text { about the line } x=4y=x4,y=0,y=1; about the line x=4

Sketch the region bounded by the graphs of the given equations and find the area of that region. y = −x2-x^{2}−x2 + 4, y = 2x + 1

Find the average value of the function on the given interval. g(x)=5xe−x2,[0,5]g(x)=5 x e^{-x^{2}},[0,5]g(x)=5xe−x2,[0,5]

Sketch the region bounded by the graphs of the given equations and find the area of that region. y = sec2 x + 6, y = 8, x = - π4\frac{\pi}{4}4π​ , x = π4\frac{\pi}{4}4π​

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations and inequalities about the y-axis. x2x^{2}x2 - y2y^{2}y2 = 16, x ≥\ge≥ 0, y = - 4, y = 4

Find the average value of the function f(t)=8tsin⁡t2f(t)=8 t \sin t^{2}f(t)=8tsint2 on the interval [0,π][0, \sqrt{\pi}][0,π​] .

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

Find the volume of the solid obtained by rotating the region bounded by x=y2x=y^{2}x=y2 and x=5yx=5 yx=5y about the y-axis.

Sketch the region enclosed by y=1+x and y=4+x4y=1+\sqrt{x} \text { and } y=\frac{4+x}{4}y=1+x​ and y=44+x​ . Find t he area of the region.

Functions and Limits

Derivatives

Applications of Differentiation

Integrals

Applications of Integration

Techniques of Integration

Further Applications 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

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

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

Find the volume of the frustum of a pyramid with square base of side $b=17$ square top of side $a=5,$ and height $h=19 .$

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 $f(m)=f_{a v e}=\frac{1}{a-b} \int_{a}^{b} f(t) d t .$

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated axis. $y=\frac{e^{x / 2}}{\left(1+e^{x}\right)^{5 / 2}}$ , $y=0$ , $x=0$ , $x=4$ ; the x-axis

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

Sketch the region bounded by the graphs of the given equations and find the area of that region. y = $-x^{2}$ + 4, y = 2x + 1

Find the average value of the function on the given interval. $g(x)=5 x e^{-x^{2}},[0,5]$

Sketch the region bounded by the graphs of the given equations and find the area of that region. y = sec² x + 6, y = 8, x = - $\frac{\pi}{4}$ , x = $\frac{\pi}{4}$

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

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

Find the volume of the solid obtained by rotating the region bounded by $x=y^{2}$ and $x=5 y$ about the y-axis.

Sketch the region enclosed by $y=1+\sqrt{x} \text { and } y=\frac{4+x}{4}$ . Find t he area of the region.