Question 1

Evaluate the iterated integral.
$$\int _ { 1 } ^ { 3 } \int _ { y } ^ { 3 } 12 x y d x d y$$

Accepted Answer

A)  96 
B)  116 
C)  106 
D)  126 
E)  101

Question 2

Use cylindrical coordinates to evaluate $\iiint _ { T } \sqrt { x ^ { 2 } + y ^ { 2 } } d V$, where $T$ is the solid bounded by the cylinder $x ^ { 2 } + y ^ { 2 } = 1$ and the planes $z = 4$ and $z = 8$.

Accepted Answer

A)  $\frac { 8 } { 3 } \pi$ 
B)  $4 \pi$ 
C)  $32 \pi$ 
D)  $48 \pi$

Question 3

Use cylindrical or spherical coordinates, whichever seems more appropriate, to evaluate $\iiint _ { E } z d V$ where $E$ lies above the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ and below the plane $z = 16$.

Accepted Answer

A)  $- 160.28$ 
B)  $176.38$ 
C)  $4,289.32$ 
D)  $175.93$ 
E)  $175.37$

Question 4

Find the area of the surface.
The part of the surface $z = 25 - x ^ { 2 } - y ^ { 2 }$ that lies above the $x y$-plane.

Accepted Answer

A)  $( \sqrt { 626 } - 626 )$ 
B)  $\frac { 2 } { 75 } ( 626 \sqrt { 626 } + 1 )$ 
C)  $\frac { 2 } { 75 } \pi ( 626 \sqrt { 626 } - 1 )$ 
D)  $\frac { 2 } { 75 } ( \sqrt { 626 } + 1 )$ 
E)  $\pi ( 626 \sqrt { 626 } - 1 )$

Question 5

Find the center of mass of the lamina that occupies the region $D$ and has the given density function, if $D$ is bounded by the parabola $y = 1 - x ^ { 2 }$ and the $x$-axis.
$$\rho ( x , y ) = 6 y$$

Accepted Answer

The answer of Find the center of mass of the...

Question 6

Evaluate the integral by making an appropriate change of variables. Round your answer to two decimal places.
$$\iint _ { z } \frac { 2 x - 3 y } { 5 x + 2 y } d A$$
$R$ is the parallelogram bounded by the lines
$$2 x - 3 y = - 5,2 x - 3 y = - 2,5 x + 2 y = - 5,5 x + 2 y = - 3 \text {. }$$

Accepted Answer

The answer of Evaluate the integral by making an appropriate...

Question 7

$$\text { Identify the surface with equation } r ^ { 2 } + z ^ { 2 } = 49 \text {. }$$

Accepted Answer

The answer of \[\text { Identify the surface with equation...

Question 8

A swimming pool is circular with a $20 - \mathrm { ft }$ diameter. The depth is constant along east-west lines and increases linearly from $3 \mathrm { ft }$ at the south end to $9 \mathrm { ft }$ at the north end. Find the volume of water in the pool.

Accepted Answer

A)  $610 \pi \mathrm { tt } ^ { 3 }$ 
B)  $800 \pi \mathrm { tt } ^ { 3 }$ 
C)  $700 \pi \mathrm { tt } ^ { 3 }$ 
D)  $900 \pi \mathrm { tt } ^ { 3 }$ 
E)  $600 \pi \mathrm { tt } ^ { 3 }$

Question 9

Find the mass and the center of mass of the lamina occupying the region $R$, where $R$ is the region bounded by the graphs of $y = \sin 3 x , y = 0 , x = 0$, and $x = \frac { \pi } { 3 }$, and having the mass density $\rho ( x , y ) = 4 y$

Accepted Answer

The answer of Find the mass and the center of...

Question 10

Find the center of mass of a lamina in the shape of an isosceles right triangle with equal sides of length $a = 15$ if the density at any point is proportional to the square of the distance from the vertex opposite the hypotenuse. Assume the vertex opposite the hypotenuse is located at $( 0,0 )$, and that the sides are along the positive axes.

Accepted Answer

The answer of Find the center of mass of a...

Question 11

Calculate the iterated integral.
$$\int _ { 0 } ^ { x } \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { \sqrt { 1 - y ^ { 2 } } } 4 y \sin x d z d y d x$$

Accepted Answer

The answer of Calculate the iterated integral.
\[\int _ { 0...

Question 12

Calculate the iterated integral. $\int _ { 0 } ^ { x } \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { \sqrt { 1 - y ^ { 2 } } } 4 y \sin x d z d y d x$

Accepted Answer

A)  $\frac { 8 } { 3 }$ 
B)  4 
C)  0 
D)  $\frac { 3 } { 8 }$ 
E)  None of these

Question 13

Calculate the iterated integral.
$$\int _ { - 1 } ^ { 1 } \int _ { 0 } ^ { 1 } 2 y e ^ { x y } d x d y$$

Accepted Answer

The answer of Calculate the iterated integral.
\[\int _ { -...

Question 14

$$\text { Use a double integral to find the area of the region } R \text { where } R \text { is bounded by the circle } r = 8 \sin \theta \text {. }$$

Accepted Answer

The answer of \[\text { Use a double integral to...

Question 15

Use spherical coordinates.
Evaluate $\iiint _ { B } \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { 2 } d V$, where $B$ is the ball with center the origin and radius 5 .

Accepted Answer

A)  $\frac { 559872 } { 7 } \pi$ 
B)  $\frac { 312500 } { 7 } \pi$ 
C)  $\frac { 4374 } { 7 } \pi$ 
D)  $\frac { 4374 } { 7 }$ 
E)  None of these

Question 16

Find the center of mass of a lamina in the shape of an isosceles right triangle with equal sides of length $a = 25$ if the density at any point is proportional to the square of the distance from the vertex opposite the hypotenuse. Assume the vertex opposite the hypotenuse is located at $( 0,0 )$, and that the sides are along the positive axes.

Accepted Answer

A)  $( 10,10 )$ 
B)  $( 10,25 )$ 
C)  $( 25,25 )$ 
D)  $( 5,10 )$ 
E)  None of these

Question 17

Use a triple integral to find the volume of the solid bounded by $x = y ^ { 2 }$ and the planes $z = 0$ and $x + z = 3$.

Accepted Answer

A)  $8.3$ 
B)  $18.3$ 
C)  $2.5$ 
D)  $15.3$ 
E)  $11.3$

Question 18

Use the Midpoint Rule with four squares of equal size to estimate the double integral.
$$\iint _ { R } \cos \left( x ^ { 4 } + y ^ { 4 } \right) d A , R = \{ ( x , y ) \mid 0 \leq x \leq 0.5,0 \leq y \leq 0.5 \}$$

Accepted Answer

A)  $0.575$ 
B)  $0.6622$ 
C)  $0.2499$ 
D)  $0.4721$ 
E)  $0.7622$

Question 19

Find the area of the part of hyperbolic paraboloid $z = y ^ { 2 } - x ^ { 2 }$ that lies between the cylinders $x ^ { 2 } + y ^ { 2 } = 1$ and $x ^ { 2 } + y ^ { 2 } = 9$.

Accepted Answer

The answer of Find the area of the part of...

Question 20

$$\text { Find the area of the part of the sphere } x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 36 z \text { that lies inside the paraboloid } z = x ^ { 2 } + y ^ { 2 } \text {. }$$

Accepted Answer

The answer of \[\text { Find the area of the...

Exam 15: Multiple Integrals

Evaluate the iterated integral. $\int _ { 1 } ^ { 3 } \int _ { y } ^ { 3 } 12 x y d x d y$

Use cylindrical coordinates to evaluate $\iiint _ { T } \sqrt { x ^ { 2 } + y ^ { 2 } } d V$ , where $T$ is the solid bounded by the cylinder $x ^ { 2 } + y ^ { 2 } = 1$ and the planes $z = 4$ and $z = 8$ .

Use cylindrical or spherical coordinates, whichever seems more appropriate, to evaluate $\iiint _ { E } z d V$ where $E$ lies above the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ and below the plane $z = 16$ .

Find the area of the surface. The part of the surface $z = 25 - x ^ { 2 } - y ^ { 2 }$ that lies above the $x y$ -plane.

Find the center of mass of the lamina that occupies the region $D$ and has the given density function, if $D$ is bounded by the parabola $y = 1 - x ^ { 2 }$ and the $x$ -axis. $\rho ( x , y ) = 6 y$

Evaluate the integral by making an appropriate change of variables. Round your answer to two decimal places. $\iint _ { z } \frac { 2 x - 3 y } { 5 x + 2 y } d A$ $R$ is the parallelogram bounded by the lines $2 x - 3 y = - 5,2 x - 3 y = - 2,5 x + 2 y = - 5,5 x + 2 y = - 3 \text {. }$

$\text { Identify the surface with equation } r ^ { 2 } + z ^ { 2 } = 49 \text {. }$

A swimming pool is circular with a $20 - \mathrm { ft }$ diameter. The depth is constant along east-west lines and increases linearly from $3 \mathrm { ft }$ at the south end to $9 \mathrm { ft }$ at the north end. Find the volume of water in the pool.

Find the mass and the center of mass of the lamina occupying the region $R$ , where $R$ is the region bounded by the graphs of $y = \sin 3 x , y = 0 , x = 0$ , and $x = \frac { \pi } { 3 }$ , and having the mass density $\rho ( x , y ) = 4 y$

Calculate the iterated integral. $\int _ { 0 } ^ { x } \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { \sqrt { 1 - y ^ { 2 } } } 4 y \sin x d z d y d x$

Calculate the iterated integral. $\int _ { 0 } ^ { x } \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { \sqrt { 1 - y ^ { 2 } } } 4 y \sin x d z d y d x$

Calculate the iterated integral. $\int _ { - 1 } ^ { 1 } \int _ { 0 } ^ { 1 } 2 y e ^ { x y } d x d y$

$\text { Use a double integral to find the area of the region } R \text { where } R \text { is bounded by the circle } r = 8 \sin \theta \text {. }$

Use spherical coordinates. Evaluate $\iiint _ { B } \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { 2 } d V$ , where $B$ is the ball with center the origin and radius 5 .

Use a triple integral to find the volume of the solid bounded by $x = y ^ { 2 }$ and the planes $z = 0$ and $x + z = 3$ .

Use the Midpoint Rule with four squares of equal size to estimate the double integral. $\iint _ { R } \cos \left( x ^ { 4 } + y ^ { 4 } \right) d A , R = \{ ( x , y ) \mid 0 \leq x \leq 0.5,0 \leq y \leq 0.5 \}$

Find the area of the part of hyperbolic paraboloid $z = y ^ { 2 } - x ^ { 2 }$ that lies between the cylinders $x ^ { 2 } + y ^ { 2 } = 1$ and $x ^ { 2 } + y ^ { 2 } = 9$ .

$\text { Find the area of the part of the sphere } x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 36 z \text { that lies inside the paraboloid } z = x ^ { 2 } + y ^ { 2 } \text {. }$

Exam 15: Multiple Integrals

Evaluate the iterated integral. ∫13∫y312xydxdy\int _ { 1 } ^ { 3 } \int _ { y } ^ { 3 } 12 x y d x d y∫13​∫y3​12xydxdy

Use cylindrical coordinates to evaluate ∭Tx2+y2dV\iiint _ { T } \sqrt { x ^ { 2 } + y ^ { 2 } } d V∭T​x2+y2​dV , where TTT is the solid bounded by the cylinder x2+y2=1x ^ { 2 } + y ^ { 2 } = 1x2+y2=1 and the planes z=4z = 4z=4 and z=8z = 8z=8 .

Use cylindrical or spherical coordinates, whichever seems more appropriate, to evaluate ∭EzdV\iiint _ { E } z d V∭E​zdV where EEE lies above the paraboloid z=x2+y2z = x ^ { 2 } + y ^ { 2 }z=x2+y2 and below the plane z=16z = 16z=16 .

Find the area of the surface. The part of the surface z=25−x2−y2z = 25 - x ^ { 2 } - y ^ { 2 }z=25−x2−y2 that lies above the xyx yxy -plane.

Find the center of mass of the lamina that occupies the region DDD and has the given density function, if DDD is bounded by the parabola y=1−x2y = 1 - x ^ { 2 }y=1−x2 and the xxx -axis. ρ(x,y)=6y\rho ( x , y ) = 6 yρ(x,y)=6y

Identify the surface with equation r2+z2=49. \text { Identify the surface with equation } r ^ { 2 } + z ^ { 2 } = 49 \text {. } Identify the surface with equation r2+z2=49.

A swimming pool is circular with a 20−ft20 - \mathrm { ft }20−ft diameter. The depth is constant along east-west lines and increases linearly from 3ft3 \mathrm { ft }3ft at the south end to 9ft9 \mathrm { ft }9ft at the north end. Find the volume of water in the pool.

Calculate the iterated integral. ∫0x∫01∫01−y24ysin⁡xdzdydx\int _ { 0 } ^ { x } \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { \sqrt { 1 - y ^ { 2 } } } 4 y \sin x d z d y d x∫0x​∫01​∫01−y2​​4ysinxdzdydx

Calculate the iterated integral. ∫0x∫01∫01−y24ysin⁡xdzdydx\int _ { 0 } ^ { x } \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { \sqrt { 1 - y ^ { 2 } } } 4 y \sin x d z d y d x∫0x​∫01​∫01−y2​​4ysinxdzdydx

Calculate the iterated integral. ∫−11∫012yexydxdy\int _ { - 1 } ^ { 1 } \int _ { 0 } ^ { 1 } 2 y e ^ { x y } d x d y∫−11​∫01​2yexydxdy

Use spherical coordinates. Evaluate ∭B(x2+y2+z2) 2dV\iiint _ { B } \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { 2 } d V∭B​(x2+y2+z2) 2dV , where BBB is the ball with center the origin and radius 5 .

Use a triple integral to find the volume of the solid bounded by x=y2x = y ^ { 2 }x=y2 and the planes z=0z = 0z=0 and x+z=3x + z = 3x+z=3 .

Find the area of the part of hyperbolic paraboloid z=y2−x2z = y ^ { 2 } - x ^ { 2 }z=y2−x2 that lies between the cylinders x2+y2=1x ^ { 2 } + y ^ { 2 } = 1x2+y2=1 and x2+y2=9x ^ { 2 } + y ^ { 2 } = 9x2+y2=9 .

Evaluate the iterated integral. $\int _ { 1 } ^ { 3 } \int _ { y } ^ { 3 } 12 x y d x d y$

Use cylindrical coordinates to evaluate $\iiint _ { T } \sqrt { x ^ { 2 } + y ^ { 2 } } d V$ , where $T$ is the solid bounded by the cylinder $x ^ { 2 } + y ^ { 2 } = 1$ and the planes $z = 4$ and $z = 8$ .

Use cylindrical or spherical coordinates, whichever seems more appropriate, to evaluate $\iiint _ { E } z d V$ where $E$ lies above the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ and below the plane $z = 16$ .

Find the area of the surface. The part of the surface $z = 25 - x ^ { 2 } - y ^ { 2 }$ that lies above the $x y$ -plane.

Find the center of mass of the lamina that occupies the region $D$ and has the given density function, if $D$ is bounded by the parabola $y = 1 - x ^ { 2 }$ and the $x$ -axis. $\rho ( x , y ) = 6 y$

$\text { Identify the surface with equation } r ^ { 2 } + z ^ { 2 } = 49 \text {. }$

A swimming pool is circular with a $20 - \mathrm { ft }$ diameter. The depth is constant along east-west lines and increases linearly from $3 \mathrm { ft }$ at the south end to $9 \mathrm { ft }$ at the north end. Find the volume of water in the pool.

Calculate the iterated integral. $\int _ { 0 } ^ { x } \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { \sqrt { 1 - y ^ { 2 } } } 4 y \sin x d z d y d x$

Calculate the iterated integral. $\int _ { 0 } ^ { x } \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { \sqrt { 1 - y ^ { 2 } } } 4 y \sin x d z d y d x$

Calculate the iterated integral. $\int _ { - 1 } ^ { 1 } \int _ { 0 } ^ { 1 } 2 y e ^ { x y } d x d y$

Use spherical coordinates. Evaluate $\iiint _ { B } \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { 2 } d V$ , where $B$ is the ball with center the origin and radius 5 .

Use a triple integral to find the volume of the solid bounded by $x = y ^ { 2 }$ and the planes $z = 0$ and $x + z = 3$ .

Find the area of the part of hyperbolic paraboloid $z = y ^ { 2 } - x ^ { 2 }$ that lies between the cylinders $x ^ { 2 } + y ^ { 2 } = 1$ and $x ^ { 2 } + y ^ { 2 } = 9$ .