Question 1

Find the fluid force of a square vertical plate submerged in water, where meters and the weight-density of water is 9800 newtons per cubic meter.

Accepted Answer

A)  $171,225,600$ newtons 
B)  $376,476,800$ newtons 
C)  $40,336,800$ newtons 
D)  $349,585,600$ newtons 
E)  $25,930,800$ newtons 
A)  $171,225,600$ newtons 
B)  $376,476,800$ newtons 
C)  $40,336,800$ newtons 
D)  $349,585,600$ newtons 
E)  $25,930,800$ newtons

Question 2

A porthole on a vertical side of a submarine (submerged in seawater) is $2 \text { square }$ $\text { feet }$ . Find the fluid force on the porthole, assuming that the center of the square is feet below the surface.

Accepted Answer

A)  $6,944 \mathrm { lb }$ 
B)  $3,472 \mathrm { lb }$ 
C)  $5,376 \mathrm { lb }$ 
D)  $5,208 \mathrm { lb }$ 
E)  $3,584 \mathrm { lb }$ 
A)  $6,944 \mathrm { lb }$ 
B)  $3,472 \mathrm { lb }$ 
C)  $5,376 \mathrm { lb }$ 
D)  $5,208 \mathrm { lb }$ 
E)  $3,584 \mathrm { lb }$

Question 3

$\text { Find the arc length from } ( 0,4 ) \text { clockwise to } ( 2,2 \sqrt { 3 } ) \text { along the circle } x ^ { 2 } + y ^ { 2 } = 16 \text {. }$ Round your answer to four decimal places.

Accepted Answer

A) 4.1888 
B) 0.2618 
C) 0.1309 
D) 1.8546 
E) 2.0944 
A) 4.1888 
B) 0.2618 
C) 0.1309 
D) 1.8546 
E) 2.0944

Question 4

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the 
$X \text {-axis. }$ $y = \frac { 1 } { \sqrt { x + 13 } } , y = 0 , x = 0 , x = 9$

Accepted Answer

A)  $\frac { 31 } { 26 } \pi$ 
B)  $\ln \left( \frac { 22 } { 13 } \right) \pi$ 
C)  $\ln \left( \frac { 31 } { 26 } \right) \pi$ 
D)  $\frac { 31 } { 13 } \pi$ 
E)  $\ln \left( \frac { 31 } { 13 } \right) \pi$ 
A)  $\frac { 31 } { 26 } \pi$ 
B)  $\ln \left( \frac { 22 } { 13 } \right) \pi$ 
C)  $\ln \left( \frac { 31 } { 26 } \right) \pi$ 
D)  $\frac { 31 } { 13 } \pi$ 
E)  $\ln \left( \frac { 31 } { 13 } \right) \pi$

Question 5

$\text { The surface of a machine part is the region between the graphs of } y _ { 1 } = | x | \text { and }$ $y _ { 2 } = 0.080 x ^ { 2 } + k$ as shown in the figure. Find $k$ if the parabola is tangent to the graph of $y _ { 1 }$. Round your answer to three decimal places.

Accepted Answer

A)  $3.125$ 
B)  $0.160$ 
C)  $0.080$ 
D)  $0.320$ 
E)  $6.250$ 
A)  $3.125$ 
B)  $0.160$ 
C)  $0.080$ 
D)  $0.320$ 
E)  $6.250$

Question 6

Find the area of the surface generated by revolving the curve about the x-axis. $$y = 4 \sqrt { x } , 1 \leq x \leq 7$$

Accepted Answer

A)  $\frac { 1 } { 3 } \left( 44 ^ { \frac { 3 } { 2 } } - 20 ^ { \frac { 3 } { 2 } } \right) \pi$ 
B)  $\frac { 2 } { 3 } \left( 32 ^ { \frac { 3 } { 2 } } - 8 ^ { \frac { 3 } { 2 } } \right) \pi$ 
C)  $\frac { 1 } { 3 } \left( 32 ^ { \frac { 3 } { 2 } } - 8 ^ { \frac { 3 } { 2 } } \right) \pi$ 
D)  $\frac { 2 } { 3 } \left( 44 ^ { \frac { 3 } { 2 } } - 20 ^ { \frac { 3 } { 2 } } \right) \pi$ 
E)  $\frac { 1 } { 6 } \left( 32 ^ { \frac { 3 } { 2 } } - 8 ^ { \frac { 3 } { 2 } } \right) \pi$ 
A)  $\frac { 1 } { 3 } \left( 44 ^ { \frac { 3 } { 2 } } - 20 ^ { \frac { 3 } { 2 } } \right) \pi$ 
B)  $\frac { 2 } { 3 } \left( 32 ^ { \frac { 3 } { 2 } } - 8 ^ { \frac { 3 } { 2 } } \right) \pi$ 
C)  $\frac { 1 } { 3 } \left( 32 ^ { \frac { 3 } { 2 } } - 8 ^ { \frac { 3 } { 2 } } \right) \pi$ 
D)  $\frac { 2 } { 3 } \left( 44 ^ { \frac { 3 } { 2 } } - 20 ^ { \frac { 3 } { 2 } } \right) \pi$ 
E)  $\frac { 1 } { 6 } \left( 32 ^ { \frac { 3 } { 2 } } - 8 ^ { \frac { 3 } { 2 } } \right) \pi$

Question 7

Use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region bounded by $y = \frac { 1 } { \sqrt { 3 \pi } } e ^ { - x ^ { 2 } / 3 } , y = 0 , x = 0$, and $x = 3$ about the $y$-axis. Round your answer to three decimal places.

Accepted Answer

A)  $5.987$ 
B)  $0.561$ 
C)  $2.047$ 
D)  $0.972$ 
E)  $2.917$ 
A)  $5.987$ 
B)  $0.561$ 
C)  $2.047$ 
D)  $0.972$ 
E)  $2.917$

Question 8

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations $y = 10 x ^ { 2 } , y = 0$, and $x = 2$ about the line $y = 40$.

Accepted Answer

A)  $\frac { 448 } { 3 }$ 
B)  $\frac { 4,480 } { 3 } \pi$ 
C)  $\frac { 448 } { 3 } \pi$ 
D)  $\frac { 4,480 } { 3 }$ 
E)  $\frac { 4,840 } { 3 } \pi$ 
A)  $\frac { 448 } { 3 }$ 
B)  $\frac { 4,480 } { 3 } \pi$ 
C)  $\frac { 448 } { 3 } \pi$ 
D)  $\frac { 4,480 } { 3 }$ 
E)  $\frac { 4,840 } { 3 } \pi$

Question 9

$\text { Find } ( \bar { x } , \bar { y } ) \text { for the lamina of uniform density } \rho \text { bounded by the graphs of the }$ equations $x = 9 - y ^ { 2 }$ and $x = 0$.

Accepted Answer

A) $( \bar { x } , \bar { y } ) = \left( 0 , \frac { 18 } { 5 } \right)$ 
B)  $( \bar { x } , \bar { y } ) = \left( \frac { 18 } { 5 } , 0 \right)$ 
C)  $( \bar { x } , \bar { y } ) = \left( 0 , \frac { 36 } { 5 } \right)$ 
D)  $( \bar { x } , \bar { y } ) = \left( \frac { 6 } { 5 } , 0 \right)$ 
E)  $( \bar { x } , \bar { y } ) = \left( \frac { 36 } { 5 } , 0 \right)$ 
A) $( \bar { x } , \bar { y } ) = \left( 0 , \frac { 18 } { 5 } \right)$ 
B)  $( \bar { x } , \bar { y } ) = \left( \frac { 18 } { 5 } , 0 \right)$ 
C)  $( \bar { x } , \bar { y } ) = \left( 0 , \frac { 36 } { 5 } \right)$ 
D)  $( \bar { x } , \bar { y } ) = \left( \frac { 6 } { 5 } , 0 \right)$ 
E)  $( \bar { x } , \bar { y } ) = \left( \frac { 36 } { 5 } , 0 \right)$

Question 10

Use the disk or the shell method to find the volume of the solid generated by revolving the region bounded by the graphs of the equations $y = 2 x ^ { 3 } , y = 0 , x = 3$ about the line $x = 8$.

Accepted Answer

A)  $\frac { 2,268 \pi } { 5 }$ 
B)  $\frac { 2,754 \pi } { 5 }$ 
C)  $\frac { 1,863 \pi } { 5 }$ 
D)  $\frac { 9,477 \pi } { 5 }$ 
E)  $\frac { 2,484 \pi } { 5 }$ 
A)  $\frac { 2,268 \pi } { 5 }$ 
B)  $\frac { 2,754 \pi } { 5 }$ 
C)  $\frac { 1,863 \pi } { 5 }$ 
D)  $\frac { 9,477 \pi } { 5 }$ 
E)  $\frac { 2,484 \pi } { 5 }$

Question 11

A tank on a water tower is a sphere of radius 65 feet. Determine the depth of the water when the tank is filled to one-fourth of its total capacity. (Note: Use the zero or root feature of a Graphing utility after evaluating the definite integral.) Round your answer to two decimal places.

Accepted Answer

A) 20.66 feet 
B) 54.63 feet 
C) 34.67 feet 
D) 42.43 feet 
E) 10.36 feet 
A) 20.66 feet 
B) 54.63 feet 
C) 34.67 feet 
D) 42.43 feet 
E) 10.36 feet

Question 12

Find the fluid force on the vertical side of the tank, where the dimensions are given in feet. Assume that the tank is full of water. Note: The density of water is 62.4 lbs per cubic foot.

Accepted Answer

A)  $93.6 \mathrm { lb }$ 
B)  $3369.6 \mathrm { lb }$ 
C)  $280.8 \mathrm { lb }$ 
D)  $561.6 \mathrm { lb }$ 
E)  $1684.8 \mathrm { lb }$ 
A)  $93.6 \mathrm { lb }$ 
B)  $3369.6 \mathrm { lb }$ 
C)  $280.8 \mathrm { lb }$ 
D)  $561.6 \mathrm { lb }$ 
E)  $1684.8 \mathrm { lb }$

Question 13

Find the area of the surface generated by revolving the curve about the y-axis. $$y = 169 - x ^ { 2 } , 0 \leq x \leq 13$$

Accepted Answer

A) $\frac { 114,245 ^ { \frac { 3 } { 2 } } - 1 } { 12 } \pi$ 
B)  $\frac { 677 ^ { \frac { 3 } { 2 } } - 1 } { 12 } \pi$ 
C) $\frac { 114,245 ^ { \frac { 3 } { 2 } } - 1 } { 6 } \pi$ 
D) $\frac { 114,245 ^ { \frac { 3 } { 2 } } - 1 } { 3 } \pi$ 
E) $\frac { 677 ^ { \frac { 3 } { 2 } } - 1 } { 6 } \pi$ 
A) $\frac { 114,245 ^ { \frac { 3 } { 2 } } - 1 } { 12 } \pi$ 
B)  $\frac { 677 ^ { \frac { 3 } { 2 } } - 1 } { 12 } \pi$ 
C) $\frac { 114,245 ^ { \frac { 3 } { 2 } } - 1 } { 6 } \pi$ 
D) $\frac { 114,245 ^ { \frac { 3 } { 2 } } - 1 } { 3 } \pi$ 
E) $\frac { 677 ^ { \frac { 3 } { 2 } } - 1 } { 6 } \pi$

Question 14

Find the center of mass of the point masses lying on the x-axis. $\begin{array} { l } 
m _ { 1 } = 10 , m _ { 2 } = 1 , m _ { 3 } = 7 \
x _ { 1 } = 3 , x _ { 2 } = 10 , x _ { 3 } = 4
\end{array}$

Accepted Answer

A)  $\bar { x } = \frac { 35 } { 9 }$ 
B)  $\bar { x } = \frac { 67 } { 18 }$ 
C)  $\bar { x } = \frac { 34 } { 9 }$ 
D)  $\bar { x } = \frac { 71 } { 21 }$ 
E)  $\bar { x } = \frac { 71 } { 18 }$ 
A)  $\bar { x } = \frac { 35 } { 9 }$ 
B)  $\bar { x } = \frac { 67 } { 18 }$ 
C)  $\bar { x } = \frac { 34 } { 9 }$ 
D)  $\bar { x } = \frac { 71 } { 21 }$ 
E)  $\bar { x } = \frac { 71 } { 18 }$

Question 15

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line $y = 2$ $$y = \frac { 1 } { 2 } x ^ { 2 } , y = 2 , x = 0$$

Accepted Answer

A)  $\frac { 8 } { 15 } \pi$ 
B)  $\frac { 32 } { 15 } \pi$ 
C)  $\frac { 64 } { 15 } \pi$ 
D)  $\frac { 4 } { 15 } \pi$ 
E)  $\frac { 16 } { 15 } \pi$ 
A)  $\frac { 8 } { 15 } \pi$ 
B)  $\frac { 32 } { 15 } \pi$ 
C)  $\frac { 64 } { 15 } \pi$ 
D)  $\frac { 4 } { 15 } \pi$ 
E)  $\frac { 16 } { 15 } \pi$

Question 16

A quantity of a gas with an initial volume of 1 cubic foot and a pressure of 2000 pounds per square foot expands to a volume of 7 cubic feet. Find the work done by the gas. Round Your answer to two decimal places.

Accepted Answer

A) 3,841.82 ft-lb 
B) 3,891.82 ft-lb 
C) 1,714.29 ft-lb 
D) 1,959.18 ft-lb 
E) 3,583.52 ft-lb 
A) 3,841.82 ft-lb 
B) 3,891.82 ft-lb 
C) 1,714.29 ft-lb 
D) 1,959.18 ft-lb 
E) 3,583.52 ft-lb

Question 17

Concrete sections for the new building have the dimensions (in meters) and shape as shown in the figure (the picture is not necessarily drawn to scale). One cubic meter of concrete Weighs 4320 pounds. Find the weight of the section. Round your answer to the nearest pound.

Accepted Answer

A)  268,492 pounds 
B)  263,654 pounds 
C)  216,267 pounds 
D)  242,187 pounds 
E)  251,865 pounds 
A)  268,492 pounds 
B)  263,654 pounds 
C)  216,267 pounds 
D)  242,187 pounds 
E)  251,865 pounds

Question 18

Find Mx, My, and ( $\bar{x}$, $\bar{y}$ ) for the lamina of uniform density $\rho$ bounded by the graphs of the equations $x = 4 - y ^ { 2 } , x = 0$.

Accepted Answer

A) $M _ { x } = 0 , M _ { y } = \frac { 256 } { 15 } \beta , \bar { x } = 0 , \bar { y } = \frac { 4 } { 5 }$ 
B)  $M _ { x } = \frac { 256 } { 15 } \rho , M _ { y } = 0 , \bar { x } = \frac { 8 } { 5 } , \bar { y } = \frac { 4 } { 5 }$ 
C)  $M _ { x } = \frac { 256 } { 15 } \alpha , M _ { y } = 0 , \bar { x } = \frac { 4 } { 5 } , \bar { y } = \frac { 4 } { 5 }$ 
D) $M _ { x } = 0 , M _ { y } = \frac { 256 } { 15 } a , \bar { y } = \frac { 8 } { 5 } , \bar { y } = 0$ 
E)  $ M _ { x } = 0 , M _ { y } = 0 , \bar { x } = 0 , \bar { y } = 0$ 
A) $M _ { x } = 0 , M _ { y } = \frac { 256 } { 15 } \beta , \bar { x } = 0 , \bar { y } = \frac { 4 } { 5 }$ 
B)  $M _ { x } = \frac { 256 } { 15 } \rho , M _ { y } = 0 , \bar { x } = \frac { 8 } { 5 } , \bar { y } = \frac { 4 } { 5 }$ 
C)  $M _ { x } = \frac { 256 } { 15 } \alpha , M _ { y } = 0 , \bar { x } = \frac { 4 } { 5 } , \bar { y } = \frac { 4 } { 5 }$ 
D) $M _ { x } = 0 , M _ { y } = \frac { 256 } { 15 } a , \bar { y } = \frac { 8 } { 5 } , \bar { y } = 0$ 
E)  $ M _ { x } = 0 , M _ { y } = 0 , \bar { x } = 0 , \bar { y } = 0$

Question 19

$\text { Suppose that } R _ { 1 } = 6.81 + 0.86 t \text { and } R _ { 2 } = 6.81 + 0.35 t \text { model the revenue (in billions }$ of dollars) for a large corporation. The model $R _ { 1 }$ gives projected annual revenues from 2008 through 2015 , with $t = 8$ corresponding to 2008 , and $R _ { 2 }$ gives projected revenues if there is a decrease in the rate of growth of corporate sales over the period. Approximate the total reduction in revenue if corporate sales are actually closer to the model $R _ { 2 }$. Round your answer to three decimal places.

Accepted Answer

A)  $\$ 3.570$ billion 
B)  $\$ 24.990$ billion 
C)  $\$ 19.763$ billion 
D)  S29.645 billion 
E)  $\$ 12.495$ billion 
A)  $\$ 3.570$ billion 
B)  $\$ 24.990$ billion 
C)  $\$ 19.763$ billion 
D)  S29.645 billion 
E)  $\$ 12.495$ billion

Question 20

$\text { Consider a beam of length } L = 12 \text { feet with a fulcrum } x \text { feet from one end as shown }$ in the figure. Two objects weighing 36 pounds and 108 pounds are placed at opposite ends of the beam. Find x (the distance between the fulcrum and the object weighing 36 pounds) such that the System is equilibrium.

Accepted Answer

A)  10 feet 
B)  11 feet 
C)  8 feet 
D)  9 feet 
E)  7 feet 
A)  10 feet 
B)  11 feet 
C)  8 feet 
D)  9 feet 
E)  7 feet

Find the fluid force of a square vertical plate submerged in water, where meters and the weight-density of water is 9800 newtons per cubic meter.

A porthole on a vertical side of a submarine (submerged in seawater) is $2 \text { square }$ $\text { feet }$ . Find the fluid force on the porthole, assuming that the center of the square is feet below the surface.

$\text { Find the arc length from } ( 0,4 ) \text { clockwise to } ( 2,2 \sqrt { 3 } ) \text { along the circle } x ^ { 2 } + y ^ { 2 } = 16 \text {. }$ Round your answer to four decimal places.

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the $X \text {-axis. }$ $y = \frac { 1 } { \sqrt { x + 13 } } , y = 0 , x = 0 , x = 9$

Find the area of the surface generated by revolving the curve about the x-axis. $y = 4 \sqrt { x } , 1 \leq x \leq 7$

Use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region bounded by $y = \frac { 1 } { \sqrt { 3 \pi } } e ^ { - x ^ { 2 } / 3 } , y = 0 , x = 0$ , and $x = 3$ about the $y$ -axis. Round your answer to three decimal places.

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations $y = 10 x ^ { 2 } , y = 0$ , and $x = 2$ about the line $y = 40$ .

$\text { Find } ( \bar { x } , \bar { y } ) \text { for the lamina of uniform density } \rho \text { bounded by the graphs of the }$ equations $x = 9 - y ^ { 2 }$ and $x = 0$ .

Use the disk or the shell method to find the volume of the solid generated by revolving the region bounded by the graphs of the equations $y = 2 x ^ { 3 } , y = 0 , x = 3$ about the line $x = 8$ .

A tank on a water tower is a sphere of radius 65 feet. Determine the depth of the water when the tank is filled to one-fourth of its total capacity. (Note: Use the zero or root feature of a Graphing utility after evaluating the definite integral.) Round your answer to two decimal places.

Find the fluid force on the vertical side of the tank, where the dimensions are given in feet. Assume that the tank is full of water. Note: The density of water is 62.4 lbs per cubic foot.

Find the area of the surface generated by revolving the curve about the y-axis. $y = 169 - x ^ { 2 } , 0 \leq x \leq 13$

Find the center of mass of the point masses lying on the x-axis. =10,=1,=7 =3,=10,=4

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line $y = 2$ $y = \frac { 1 } { 2 } x ^ { 2 } , y = 2 , x = 0$

A quantity of a gas with an initial volume of 1 cubic foot and a pressure of 2000 pounds per square foot expands to a volume of 7 cubic feet. Find the work done by the gas. Round Your answer to two decimal places.

Concrete sections for the new building have the dimensions (in meters) and shape as shown in the figure (the picture is not necessarily drawn to scale). One cubic meter of concrete Weighs 4320 pounds. Find the weight of the section. Round your answer to the nearest pound.

Find M_x, M_y, and ( $\bar{x}$ , $\bar{y}$ ) for the lamina of uniform density $\rho$ bounded by the graphs of the equations $x = 4 - y ^ { 2 } , x = 0$ .

Graphs and Models

A Preview of Calculus

The Derivative and the Tangent Line Problem

Extrema on an Interval

Antiderivatives and Indefinite Integration

Slope Fields and Eulers Method

Basic Integration Rules

Sequences

Conics and Calculus

Vectors in the Plane

Vector-Valued Functions

Introduction to Functions of Several Variables

Iterated Integrals and Area in the Plane

Vector Fields

Exact First-Order Equations

Filters

Exam 7: Area of a Region Between Two Curves

Find the fluid force of a square vertical plate submerged in water, where meters and the weight-density of water is 9800 newtons per cubic meter.

A porthole on a vertical side of a submarine (submerged in seawater) is 2 square 2 \text { square }2 square feet \text { feet } feet . Find the fluid force on the porthole, assuming that the center of the square is feet below the surface.

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the X-axis. X \text {-axis. }X-axis. y=1x+13,y=0,x=0,x=9y = \frac { 1 } { \sqrt { x + 13 } } , y = 0 , x = 0 , x = 9y=x+13​1​,y=0,x=0,x=9

Find the area of the surface generated by revolving the curve about the x-axis. y=4x,1≤x≤7y = 4 \sqrt { x } , 1 \leq x \leq 7y=4x​,1≤x≤7

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations y=10x2,y=0y = 10 x ^ { 2 } , y = 0y=10x2,y=0 , and x=2x = 2x=2 about the line y=40y = 40y=40 .

Use the disk or the shell method to find the volume of the solid generated by revolving the region bounded by the graphs of the equations y=2x3,y=0,x=3y = 2 x ^ { 3 } , y = 0 , x = 3y=2x3,y=0,x=3 about the line x=8x = 8x=8 .

A tank on a water tower is a sphere of radius 65 feet. Determine the depth of the water when the tank is filled to one-fourth of its total capacity. (Note: Use the zero or root feature of a Graphing utility after evaluating the definite integral.) Round your answer to two decimal places.

Find the fluid force on the vertical side of the tank, where the dimensions are given in feet. Assume that the tank is full of water. Note: The density of water is 62.4 lbs per cubic foot.

Find the area of the surface generated by revolving the curve about the y-axis. y=169−x2,0≤x≤13y = 169 - x ^ { 2 } , 0 \leq x \leq 13y=169−x2,0≤x≤13

Find the center of mass of the point masses lying on the x-axis. =10,=1,=7 =3,=10,=4

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line y=2y = 2y=2 y=12x2,y=2,x=0y = \frac { 1 } { 2 } x ^ { 2 } , y = 2 , x = 0y=21​x2,y=2,x=0

A quantity of a gas with an initial volume of 1 cubic foot and a pressure of 2000 pounds per square foot expands to a volume of 7 cubic feet. Find the work done by the gas. Round Your answer to two decimal places.

Concrete sections for the new building have the dimensions (in meters) and shape as shown in the figure (the picture is not necessarily drawn to scale). One cubic meter of concrete Weighs 4320 pounds. Find the weight of the section. Round your answer to the nearest pound.

Find Mx, My, and ( xˉ\bar{x}xˉ , yˉ\bar{y}yˉ​ ) for the lamina of uniform density ρ\rhoρ bounded by the graphs of the equations x=4−y2,x=0x = 4 - y ^ { 2 } , x = 0x=4−y2,x=0 .

Graphs and Models

A Preview of Calculus

The Derivative and the Tangent Line Problem

Extrema on an Interval

Antiderivatives and Indefinite Integration

Slope Fields and Eulers Method

Basic Integration Rules

Sequences

Conics and Calculus

Vectors in the Plane

Vector-Valued Functions

Introduction to Functions of Several Variables

Iterated Integrals and Area in the Plane

Vector Fields

Exact First-Order Equations

Filters

A porthole on a vertical side of a submarine (submerged in seawater) is $2 \text { square }$ $\text { feet }$ . Find the fluid force on the porthole, assuming that the center of the square is feet below the surface.

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the $X \text {-axis. }$ $y = \frac { 1 } { \sqrt { x + 13 } } , y = 0 , x = 0 , x = 9$

Find the area of the surface generated by revolving the curve about the x-axis. $y = 4 \sqrt { x } , 1 \leq x \leq 7$

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations $y = 10 x ^ { 2 } , y = 0$ , and $x = 2$ about the line $y = 40$ .

Use the disk or the shell method to find the volume of the solid generated by revolving the region bounded by the graphs of the equations $y = 2 x ^ { 3 } , y = 0 , x = 3$ about the line $x = 8$ .

Find the area of the surface generated by revolving the curve about the y-axis. $y = 169 - x ^ { 2 } , 0 \leq x \leq 13$

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line $y = 2$ $y = \frac { 1 } { 2 } x ^ { 2 } , y = 2 , x = 0$

Find M_x, M_y, and ( $\bar{x}$ , $\bar{y}$ ) for the lamina of uniform density $\rho$ bounded by the graphs of the equations $x = 4 - y ^ { 2 } , x = 0$ .