Question 1

Find the center of mass of the solid bounded by $z = 196 - x ^ { 2 } - y ^ { 2 } \text { and } z = 0 \text { with }$ density function $p = k z$.

Accepted Answer

A)  $( 102,102,102 )$ 
B)  $( 0,0,102 )$ 
C)  $( 0,0,98 )$ 
D)  $( 98,98,98 )$ 
E)  $( 196,0,196 )$ 
A)  $( 102,102,102 )$ 
B)  $( 0,0,102 )$ 
C)  $( 0,0,98 )$ 
D)  $( 98,98,98 )$ 
E)  $( 196,0,196 )$

Question 2

Find the center of mass of the rectangular lamina with vertices $( 0,0 ) , ( 6,0 ) , ( 0,24 )$ and $( 6,24 )$ for the density $\rho = k x y$.

Accepted Answer

A)  $( 16,4 )$ 
B)  $( 6,24 )$ 
C)  $( 0,16 )$ 
D)  $( 4,16 )$ 
E)  $( 4,8 )$ 
A)  $( 16,4 )$ 
B)  $( 6,24 )$ 
C)  $( 0,16 )$ 
D)  $( 4,16 )$ 
E)  $( 4,8 )$

Question 3

Evaluate the iterated integral below. Note that it is necessary to switch the order of integration. $\int _ { 0 } ^ { 1 } \int _ { x } ^ {1 }  e^{- 9 y ^ { 2 }} d y d x$

Accepted Answer

A)  $\frac { 1 - e ^ { 9 } } { 9 }$ 
B)  $\frac { 1 - e ^ { - 9 } } { 18 }$ 
C)  $\frac { 1 - e ^ { - 9 } } { 9 }$ 
D)  $\frac { e ^ { - 9 } - 1 } { 18 }$ 
E)  $\frac { 1 - e ^ { - 9 } } { 36 }$ 
A)  $\frac { 1 - e ^ { 9 } } { 9 }$ 
B)  $\frac { 1 - e ^ { - 9 } } { 18 }$ 
C)  $\frac { 1 - e ^ { - 9 } } { 9 }$ 
D)  $\frac { e ^ { - 9 } - 1 } { 18 }$ 
E)  $\frac { 1 - e ^ { - 9 } } { 36 }$

Question 4

Use a double integral to find the volume of the indicated solid.  
$$3 x + y + z = 3 , x > 0 , y > 0 , z > 0$$

Accepted Answer

A)  $\frac { 17 } { 26 }$ 
B)  $\frac { 26 } { 17 }$ 
C)  $\frac { 2 } { 3 }$ 
D)  $\frac { 3 } { 17 }$ 
E)  $\frac { 3 } { 2 }$ 
A)  $\frac { 17 } { 26 }$ 
B)  $\frac { 26 } { 17 }$ 
C)  $\frac { 2 } { 3 }$ 
D)  $\frac { 3 } { 17 }$ 
E)  $\frac { 3 } { 2 }$

Question 5

Find the mass of the lamina bounded by the graphs of the equations $y = e ^ { - x } , y = 0 , x = 0$, and $x = 6$ for the density $\rho = k y ^ { 2 }$

Accepted Answer

A)  $\frac { 1 } { 9 } \left( 1 - e ^ { 12 } \right) k$ 
B)  $\frac { 1 } { 3 } \left( 1 - e ^ { 18 } \right) k$ 
C)  $\frac { 1 } { 9 } \left( - 1 + e ^ { - 18 } \right) k$ 
D)  $\frac { 1 } { 3 } \left( - 1 + e ^ { - 12 } \right) k$ 
E)  $\frac { 1 } { 9 } \left( 1 - e ^ { - 18 } \right) k$ 
A)  $\frac { 1 } { 9 } \left( 1 - e ^ { 12 } \right) k$ 
B)  $\frac { 1 } { 3 } \left( 1 - e ^ { 18 } \right) k$ 
C)  $\frac { 1 } { 9 } \left( - 1 + e ^ { - 18 } \right) k$ 
D)  $\frac { 1 } { 3 } \left( - 1 + e ^ { - 12 } \right) k$ 
E)  $\frac { 1 } { 9 } \left( 1 - e ^ { - 18 } \right) k$

Question 6

Set up a double integral that gives the area of the surface of the graph of f over the region R. $f ( x , y ) = e ^ { 9 x y }$
$R = \{ ( x , y ) : 0 \leq x \leq 6,0 \leq y \leq 2 \}$

Accepted Answer

A)  $\int _ { 0 } ^ { 2 } \int _ { 0 } ^ { 6 } \sqrt { 1 + 81 e ^ { 18 x y } \left( x ^ { 2 } + y ^ { 2 } \right) } d x d y$ 
B) $\int_{0}^{2} \int_{0}^{6} \sqrt{1+81 e^{18 x y}\left(x^{2}+y^{2}\right)} d y d x$ 
C) $\int _ { 0 } ^ { 2 } \int _ { 0 } ^ { 6 } \sqrt { 1 + 81 e ^ { 9 x y } \left( x ^ { 2 } + y ^ { 2 } \right) } d x d y$ 
D)  $\int _ { 0 } ^ { 2 } \int _ { 0 } ^ { 6 } \sqrt { 1 + 9 e ^ { 9 x y } \left( x ^ { 2 } + y ^ { 2 } \right) } d x d y$ 
E)  $\int _ { 0 } ^ { 6 } \int _ { 0 } ^ { 2 } \sqrt { 1 + 81 e ^ { 18 x y } \left( x ^ { 2 } + y ^ { 2 } \right) } d x d y$ 
A)  $\int _ { 0 } ^ { 2 } \int _ { 0 } ^ { 6 } \sqrt { 1 + 81 e ^ { 18 x y } \left( x ^ { 2 } + y ^ { 2 } \right) } d x d y$ 
B) $\int_{0}^{2} \int_{0}^{6} \sqrt{1+81 e^{18 x y}\left(x^{2}+y^{2}\right)} d y d x$ 
C) $\int _ { 0 } ^ { 2 } \int _ { 0 } ^ { 6 } \sqrt { 1 + 81 e ^ { 9 x y } \left( x ^ { 2 } + y ^ { 2 } \right) } d x d y$ 
D)  $\int _ { 0 } ^ { 2 } \int _ { 0 } ^ { 6 } \sqrt { 1 + 9 e ^ { 9 x y } \left( x ^ { 2 } + y ^ { 2 } \right) } d x d y$ 
E)  $\int _ { 0 } ^ { 6 } \int _ { 0 } ^ { 2 } \sqrt { 1 + 81 e ^ { 18 x y } \left( x ^ { 2 } + y ^ { 2 } \right) } d x d y$

Question 7

Set up and evaluate a double integral to find the volume of the solid bounded by the graphs of the equations $x ^ { 2 } + z ^ { 2 } = 144$ and $y ^ { 2 } + z ^ { 2 } = 144$ in the first octant.

Accepted Answer

A)  5,184 
B)  576 
C)  3,456 
D)  1,728 
E)  1,152 
A)  5,184 
B)  576 
C)  3,456 
D)  1,728 
E)  1,152

Question 8

Find the center of mass of the lamina bounded by the graphs of the equations $x = 64 - y ^ { 2 }$ and $x = 0$ for the density $\rho = k x$

Accepted Answer

A)  $\left( \frac { 128 } { 7 } , 0 \right)$ 
B)  $\left( 0 , \frac { 128 } { 7 } \right)$ 
C)  $\left( \frac { 32 } { 7 } , 0 \right)$ 
D)  $\left( 0 , \frac { 256 } { 7 } \right)$ 
E)  $\left( \frac { 256 } { 7 } , 0 \right)$ 
A)  $\left( \frac { 128 } { 7 } , 0 \right)$ 
B)  $\left( 0 , \frac { 128 } { 7 } \right)$ 
C)  $\left( \frac { 32 } { 7 } , 0 \right)$ 
D)  $\left( 0 , \frac { 256 } { 7 } \right)$ 
E)  $\left( \frac { 256 } { 7 } , 0 \right)$

Question 9

Find the area of the portion of the surface $f ( x , y ) = 7 + \frac { 2 } { 3 } y ^ { \frac { 3 } { 2 } }$
that lies above the region $R = \{ ( x , y ) : 0 \leq x \leq 4,0 \leq y \leq 4 - x \}$. Round your answer to two decimal places.

Accepted Answer

A)  $0.05$ 
B)  $2.18$ 
C)  $11.97$ 
D)  $1.22$ 
E)  $12.51$ 
A)  $0.05$ 
B)  $2.18$ 
C)  $11.97$ 
D)  $1.22$ 
E)  $12.51$

Question 10

Evaluate the double integral below. $$\int _ { 0 } ^ { \frac { \pi } { 5 } } \int _ { 0 } ^ { 5 } r ^ { 5 } \sin \theta \cos \theta d r d \theta$$

Accepted Answer

A)  $\frac { 1202 } { 1203 } \sin \frac { \pi } { 5 }$ 
B)  $\frac { 15625 } { 12 } \sin \frac { \pi } { 5 }$ 
C)  $\frac { 12 } { 15625 } \sin ^ { 2 } \frac { \pi } { 5 }$ 
D)  $\frac { 15625 } { 12 } \sin ^ { 2 } \frac { \pi } { 5 }$ 
E)  $\frac { 1 } { 12 } \sin ^ { 2 } \frac { \pi } { 5 }$ 
A)  $\frac { 1202 } { 1203 } \sin \frac { \pi } { 5 }$ 
B)  $\frac { 15625 } { 12 } \sin \frac { \pi } { 5 }$ 
C)  $\frac { 12 } { 15625 } \sin ^ { 2 } \frac { \pi } { 5 }$ 
D)  $\frac { 15625 } { 12 } \sin ^ { 2 } \frac { \pi } { 5 }$ 
E)  $\frac { 1 } { 12 } \sin ^ { 2 } \frac { \pi } { 5 }$

Question 11

Use a double integral to find the volume of the indicated solid. $z = 16 - y ^ { 2 } , z > 0 , x > 0,3 x < y < 4$

Accepted Answer

A)  $\frac { 64 } { 3 }$ 
B)  $\frac { 254 } { 11 }$ 
C)  $\frac { 11 } { 254 }$ 
D)  $\frac { 3 } { 64 }$ 
E)  none of the above 
A)  $\frac { 64 } { 3 }$ 
B)  $\frac { 254 } { 11 }$ 
C)  $\frac { 11 } { 254 }$ 
D)  $\frac { 3 } { 64 }$ 
E)  none of the above

Question 12

Set up a triple integral for the volume of the solid bounded by $z = 7 - x ^ { 2 } - y ^ { 2 } \text { and }$ $z = 0$

Accepted Answer

A) 
$ \int_{-\sqrt{7}}^{\sqrt{7}} \int_{-7}^{\sqrt{7-x^{2}}} \int_{0}^{7-x^{2}-y^{2}} \int_{0} d z d x d y $ 
B)  
$ \int_{-7}^{7} \int_{-7}^{7-x^{2}-y^{2}} \int_{0} ^{7-{x^2}-{y^2}} d z d x d y $ 
C)  
$ \int_{-\sqrt{7}}^{\sqrt{7}} \int_{-\sqrt{7-{x^2}}}^{\sqrt{7-x^{2}}} \int_{0} ^{\sqrt{7-{x^2}-{y^2}}} d z d x d y $ 
D) 
$ \int_{-\sqrt{7}}^{\sqrt{7}} \int_{-\sqrt{7-{x^2}}}^{\sqrt{7-x^{2}}} \int_{0} ^{{7-{x^2}-{y^2}}} d z d x d y $ 
E) 
$ \int_{-7}^{7} \int_{-\sqrt{7-{x^2}}}^{\sqrt{7-x^{2}}} \int_{0} ^{{7-{x^2}-{y^2}}} d z d x d y $ 
A) 
$ \int_{-\sqrt{7}}^{\sqrt{7}} \int_{-7}^{\sqrt{7-x^{2}}} \int_{0}^{7-x^{2}-y^{2}} \int_{0} d z d x d y $ 
B)  
$ \int_{-7}^{7} \int_{-7}^{7-x^{2}-y^{2}} \int_{0} ^{7-{x^2}-{y^2}} d z d x d y $ 
C)  
$ \int_{-\sqrt{7}}^{\sqrt{7}} \int_{-\sqrt{7-{x^2}}}^{\sqrt{7-x^{2}}} \int_{0} ^{\sqrt{7-{x^2}-{y^2}}} d z d x d y $ 
D) 
$ \int_{-\sqrt{7}}^{\sqrt{7}} \int_{-\sqrt{7-{x^2}}}^{\sqrt{7-x^{2}}} \int_{0} ^{{7-{x^2}-{y^2}}} d z d x d y $ 
E) 
$ \int_{-7}^{7} \int_{-\sqrt{7-{x^2}}}^{\sqrt{7-x^{2}}} \int_{0} ^{{7-{x^2}-{y^2}}} d z d x d y $

Question 13

Find the area of the surface for the portion of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 225 \text { inside }$the cylinder $x ^ { 2 } + y ^ { 2 } = 144$

Accepted Answer

A)  $540 \pi$ 
B)  $720 \pi$ 
C)  $360 \pi$ 
D)  $60 \pi$ 
E)  $40 \pi$ 
A)  $540 \pi$ 
B)  $720 \pi$ 
C)  $360 \pi$ 
D)  $60 \pi$ 
E)  $40 \pi$

Question 14

Use a double integral to find the volume of the indicated solid.

Accepted Answer

A)  16 
B)  9 
C)  4 
D)  20 
E)  6 
A)  16 
B)  9 
C)  4 
D)  20 
E)  6

Question 15

Use the indicated change of variables to evaluate the following double integral. $$x = \frac { 1 } { 2 } ( u + v ) , y = - \frac { 1 } { 2 } ( u - v )$$
$\iint_{R} 60 x y d A$

Accepted Answer

A)  128 
B)  120 
C)  $- 120$ 
D)  118 
E)  60 
A)  128 
B)  120 
C)  $- 120$ 
D)  118 
E)  60

Question 16

Use a change of variables to find the volume of the solid region lying below the surface $z = \frac { x y } { 7 + x ^ { 2 } y ^ { 2 } }$ and above the plane region $R$ : region bounded by the graphs of $x y = 1 , x y = 4 , x = 1 , x = 2$ (Hint: Let $x = u , y = v / u$.) Round your answer to two decimal places.

Accepted Answer

A)  $0.53$ 
B)  $5.21$ 
C)  $0.37$ 
D)  $0.53$. 
E)  $10.43$. 
A)  $0.53$ 
B)  $5.21$ 
C)  $0.37$ 
D)  $0.53$. 
E)  $10.43$.

Question 17

Set up the double integral required to find the moment of inertia I, about the line $y = 1$, of the lamina bounded by the graphs of the equations $y = 9 - x ^ { 2 }$ and $y = 0$ for the density $\rho = k$. Use a computer algebra system to evaluate the double integral.

Accepted Answer

A)  $\frac { 15,606 k } { 35 }$ 
B)  $\frac { 15,426 k } { 35 }$ 
C)  $\frac { 16,326 k } { 35 }$ 
D)  $\frac { 16,236 k } { 35 }$ 
E)  $\frac { 15,516 k } { 35 }$ 
A)  $\frac { 15,606 k } { 35 }$ 
B)  $\frac { 15,426 k } { 35 }$ 
C)  $\frac { 16,326 k } { 35 }$ 
D)  $\frac { 16,236 k } { 35 }$ 
E)  $\frac { 15,516 k } { 35 }$

Question 18

Evaluate the iterated integral  $\int_{0}^{\frac{x}{2}}\int_{0} ^ {\frac{y}{7}}\int_{0} ^ {\frac{2}{y}} (3 sin y)   dx dy dz $

Accepted Answer

A)  $\frac { 3 } { 14 }$ 
B)  $\frac { 6 } { 7 }$ 
C)  $\frac { 2 } { 7 } \pi$ 
D)  $\frac { 3 } { 7 } \pi$ 
E)  $\frac { 4 } { 7 }$ 
A)  $\frac { 3 } { 14 }$ 
B)  $\frac { 6 } { 7 }$ 
C)  $\frac { 2 } { 7 } \pi$ 
D)  $\frac { 3 } { 7 } \pi$ 
E)  $\frac { 4 } { 7 }$

Question 19

Set up and evaluate a double integral to find the volume of the solid bounded by the graphs of the equations $y = 0 , z = 0 , y = x , z = x , x = 0 \text {, and } x = 33$

Accepted Answer

A) 1,089 
B) 35,937 
C) 35,949 
D) 11,979 
E) 2,178 
A) 1,089 
B) 35,937 
C) 35,949 
D) 11,979 
E) 2,178

Question 20

Evaluate the following iterated integral. $\int_{1}^{2}\int_{0} ^ {3}\int_{0} ^ {x}(2ze^{-x^2} )d x d y d z $

Accepted Answer

A)  $2 \left( 1 - e ^ { - 3 } \right)$ 
B)  $\frac { 3 } { 2 } \left( 1 - e ^ { - 9 } \right)$ 
C)  $2 \left( 1 - e ^ { - 9 } \right)$ 
D)  $\frac { 3 } { 2 } \left( e ^ { 9 } - 1 \right)$ 
E)  None of the above 
A)  $2 \left( 1 - e ^ { - 3 } \right)$ 
B)  $\frac { 3 } { 2 } \left( 1 - e ^ { - 9 } \right)$ 
C)  $2 \left( 1 - e ^ { - 9 } \right)$ 
D)  $\frac { 3 } { 2 } \left( e ^ { 9 } - 1 \right)$ 
E)  None of the above

Find the center of mass of the solid bounded by $z = 196 - x ^ { 2 } - y ^ { 2 } \text { and } z = 0 \text { with }$ density function $p = k z$ .

Find the center of mass of the rectangular lamina with vertices $( 0,0 ) , ( 6,0 ) , ( 0,24 )$ and $( 6,24 )$ for the density $\rho = k x y$ .

Evaluate the iterated integral below. Note that it is necessary to switch the order of integration. $\int _ { 0 } ^ { 1 } \int _ { x } ^ {1 } e^{- 9 y ^ { 2 }} d y d x$

Use a double integral to find the volume of the indicated solid. $3 x + y + z = 3 , x > 0 , y > 0 , z > 0$

Find the mass of the lamina bounded by the graphs of the equations $y = e ^ { - x } , y = 0 , x = 0$ , and $x = 6$ for the density $\rho = k y ^ { 2 }$

Set up a double integral that gives the area of the surface of the graph of f over the region R. $f ( x , y ) = e ^ { 9 x y }$ $R = \{ ( x , y ) : 0 \leq x \leq 6,0 \leq y \leq 2 \}$

Set up and evaluate a double integral to find the volume of the solid bounded by the graphs of the equations $x ^ { 2 } + z ^ { 2 } = 144$ and $y ^ { 2 } + z ^ { 2 } = 144$ in the first octant.

Find the center of mass of the lamina bounded by the graphs of the equations $x = 64 - y ^ { 2 }$ and $x = 0$ for the density $\rho = k x$

Find the area of the portion of the surface $f ( x , y ) = 7 + \frac { 2 } { 3 } y ^ { \frac { 3 } { 2 } }$ that lies above the region $R = \{ ( x , y ) : 0 \leq x \leq 4,0 \leq y \leq 4 - x \}$ . Round your answer to two decimal places.

Evaluate the double integral below. $\int _ { 0 } ^ { \frac { \pi } { 5 } } \int _ { 0 } ^ { 5 } r ^ { 5 } \sin \theta \cos \theta d r d \theta$

Use a double integral to find the volume of the indicated solid. $Use a double integral to find the volume of the indicated solid. z = 16 - y ^ { 2 } , z > 0 , x > 0,3 x < y < 4$ $z = 16 - y ^ { 2 } , z > 0 , x > 0,3 x < y < 4$

Set up a triple integral for the volume of the solid bounded by $z = 7 - x ^ { 2 } - y ^ { 2 } \text { and }$ $z = 0$

Find the area of the surface for the portion of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 225 \text { inside }$ the cylinder $x ^ { 2 } + y ^ { 2 } = 144$

Use a double integral to find the volume of the indicated solid.

Set up the double integral required to find the moment of inertia I, about the line $y = 1$ , of the lamina bounded by the graphs of the equations $y = 9 - x ^ { 2 }$ and $y = 0$ for the density $\rho = k$ . Use a computer algebra system to evaluate the double integral.

Evaluate the iterated integral $\int_{0}^{\frac{x}{2}}\int_{0} ^ {\frac{y}{7}}\int_{0} ^ {\frac{2}{y}} (3 sin y) dx dy dz$

Set up and evaluate a double integral to find the volume of the solid bounded by the graphs of the equations $y = 0 , z = 0 , y = x , z = x , x = 0 \text {, and } x = 33$

Evaluate the following iterated integral. $\int_{1}^{2}\int_{0} ^ {3}\int_{0} ^ {x}(2ze^{-x^2} )d x d y d z$

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

Area of a Region Between Two Curves

Basic Integration Rules

Sequences

Conics and Calculus

Vectors in the Plane

Vector-Valued Functions

Introduction to Functions of Several Variables

Vector Fields

Exact First-Order Equations

Filters

Exam 14: Iterated Integrals and Area in the Plane

Find the center of mass of the solid bounded by z=196−x2−y2 and z=0 with z = 196 - x ^ { 2 } - y ^ { 2 } \text { and } z = 0 \text { with }z=196−x2−y2 and z=0 with density function p=kzp = k zp=kz .

Find the center of mass of the rectangular lamina with vertices (0,0),(6,0),(0,24)( 0,0 ) , ( 6,0 ) , ( 0,24 )(0,0),(6,0),(0,24) and (6,24)( 6,24 )(6,24) for the density ρ=kxy\rho = k x yρ=kxy .

Evaluate the iterated integral below. Note that it is necessary to switch the order of integration. ∫01∫x1e−9y2dydx\int _ { 0 } ^ { 1 } \int _ { x } ^ {1 } e^{- 9 y ^ { 2 }} d y d x∫01​∫x1​e−9y2dydx

Use a double integral to find the volume of the indicated solid. 3x+y+z=3,x>0,y>0,z>03 x + y + z = 3 , x > 0 , y > 0 , z > 03x+y+z=3,x>0,y>0,z>0

Find the mass of the lamina bounded by the graphs of the equations y=e−x,y=0,x=0y = e ^ { - x } , y = 0 , x = 0y=e−x,y=0,x=0 , and x=6x = 6x=6 for the density ρ=ky2\rho = k y ^ { 2 }ρ=ky2

Set up a double integral that gives the area of the surface of the graph of f over the region R. f(x,y)=e9xyf ( x , y ) = e ^ { 9 x y }f(x,y)=e9xy R={(x,y):0≤x≤6,0≤y≤2}R = \{ ( x , y ) : 0 \leq x \leq 6,0 \leq y \leq 2 \}R={(x,y):0≤x≤6,0≤y≤2}

Set up and evaluate a double integral to find the volume of the solid bounded by the graphs of the equations x2+z2=144x ^ { 2 } + z ^ { 2 } = 144x2+z2=144 and y2+z2=144y ^ { 2 } + z ^ { 2 } = 144y2+z2=144 in the first octant.

Find the center of mass of the lamina bounded by the graphs of the equations x=64−y2x = 64 - y ^ { 2 }x=64−y2 and x=0x = 0x=0 for the density ρ=kx\rho = k xρ=kx

Evaluate the double integral below. ∫0π5∫05r5sin⁡θcos⁡θdrdθ\int _ { 0 } ^ { \frac { \pi } { 5 } } \int _ { 0 } ^ { 5 } r ^ { 5 } \sin \theta \cos \theta d r d \theta∫05π​​∫05​r5sinθcosθdrdθ

Use a double integral to find the volume of the indicated solid. z=16−y2,z>0,x>0,3x<y<4z = 16 - y ^ { 2 } , z > 0 , x > 0,3 x < y < 4z=16−y2,z>0,x>0,3x<y<4

Set up a triple integral for the volume of the solid bounded by z=7−x2−y2 and z = 7 - x ^ { 2 } - y ^ { 2 } \text { and }z=7−x2−y2 and z=0z = 0z=0

Find the area of the surface for the portion of the sphere x2+y2+z2=225 inside x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 225 \text { inside }x2+y2+z2=225 inside the cylinder x2+y2=144x ^ { 2 } + y ^ { 2 } = 144x2+y2=144

Use a double integral to find the volume of the indicated solid.

Use the indicated change of variables to evaluate the following double integral. x=12(u+v),y=−12(u−v)x = \frac { 1 } { 2 } ( u + v ) , y = - \frac { 1 } { 2 } ( u - v )x=21​(u+v),y=−21​(u−v) ∬R60xydA\iint_{R} 60 x y d A∬R​60xydA

Set up the double integral required to find the moment of inertia I, about the line y=1y = 1y=1 , of the lamina bounded by the graphs of the equations y=9−x2y = 9 - x ^ { 2 }y=9−x2 and y=0y = 0y=0 for the density ρ=k\rho = kρ=k . Use a computer algebra system to evaluate the double integral.

Evaluate the iterated integral ∫0x2∫0y7∫02y(3siny)dxdydz\int_{0}^{\frac{x}{2}}\int_{0} ^ {\frac{y}{7}}\int_{0} ^ {\frac{2}{y}} (3 sin y) dx dy dz ∫02x​​∫07y​​∫0y2​​(3siny)dxdydz

Set up and evaluate a double integral to find the volume of the solid bounded by the graphs of the equations y=0,z=0,y=x,z=x,x=0, and x=33y = 0 , z = 0 , y = x , z = x , x = 0 \text {, and } x = 33y=0,z=0,y=x,z=x,x=0, and x=33

Evaluate the following iterated integral. ∫12∫03∫0x(2ze−x2)dxdydz\int_{1}^{2}\int_{0} ^ {3}\int_{0} ^ {x}(2ze^{-x^2} )d x d y d z ∫12​∫03​∫0x​(2ze−x2)dxdydz

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

Area of a Region Between Two Curves

Basic Integration Rules

Sequences

Conics and Calculus

Vectors in the Plane

Vector-Valued Functions

Introduction to Functions of Several Variables

Vector Fields

Exact First-Order Equations

Filters

Find the center of mass of the solid bounded by $z = 196 - x ^ { 2 } - y ^ { 2 } \text { and } z = 0 \text { with }$ density function $p = k z$ .

Find the center of mass of the rectangular lamina with vertices $( 0,0 ) , ( 6,0 ) , ( 0,24 )$ and $( 6,24 )$ for the density $\rho = k x y$ .

Evaluate the iterated integral below. Note that it is necessary to switch the order of integration. $\int _ { 0 } ^ { 1 } \int _ { x } ^ {1 } e^{- 9 y ^ { 2 }} d y d x$

Use a double integral to find the volume of the indicated solid. $3 x + y + z = 3 , x > 0 , y > 0 , z > 0$

Find the mass of the lamina bounded by the graphs of the equations $y = e ^ { - x } , y = 0 , x = 0$ , and $x = 6$ for the density $\rho = k y ^ { 2 }$

Set up a double integral that gives the area of the surface of the graph of f over the region R. $f ( x , y ) = e ^ { 9 x y }$ $R = \{ ( x , y ) : 0 \leq x \leq 6,0 \leq y \leq 2 \}$

Set up and evaluate a double integral to find the volume of the solid bounded by the graphs of the equations $x ^ { 2 } + z ^ { 2 } = 144$ and $y ^ { 2 } + z ^ { 2 } = 144$ in the first octant.

Find the center of mass of the lamina bounded by the graphs of the equations $x = 64 - y ^ { 2 }$ and $x = 0$ for the density $\rho = k x$

Evaluate the double integral below. $\int _ { 0 } ^ { \frac { \pi } { 5 } } \int _ { 0 } ^ { 5 } r ^ { 5 } \sin \theta \cos \theta d r d \theta$

Use a double integral to find the volume of the indicated solid. $Use a double integral to find the volume of the indicated solid. z = 16 - y ^ { 2 } , z > 0 , x > 0,3 x < y < 4$ $z = 16 - y ^ { 2 } , z > 0 , x > 0,3 x < y < 4$

Set up a triple integral for the volume of the solid bounded by $z = 7 - x ^ { 2 } - y ^ { 2 } \text { and }$ $z = 0$

Find the area of the surface for the portion of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 225 \text { inside }$ the cylinder $x ^ { 2 } + y ^ { 2 } = 144$

Set up the double integral required to find the moment of inertia I, about the line $y = 1$ , of the lamina bounded by the graphs of the equations $y = 9 - x ^ { 2 }$ and $y = 0$ for the density $\rho = k$ . Use a computer algebra system to evaluate the double integral.

Evaluate the iterated integral $\int_{0}^{\frac{x}{2}}\int_{0} ^ {\frac{y}{7}}\int_{0} ^ {\frac{2}{y}} (3 sin y) dx dy dz$

Set up and evaluate a double integral to find the volume of the solid bounded by the graphs of the equations $y = 0 , z = 0 , y = x , z = x , x = 0 \text {, and } x = 33$

Evaluate the following iterated integral. $\int_{1}^{2}\int_{0} ^ {3}\int_{0} ^ {x}(2ze^{-x^2} )d x d y d z$