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 
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)  $48 \pi$ 
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)  $175.37$ 
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)  $\pi ( 626 \sqrt { 626 } - 1 )$ 
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

Sphere wit

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)  $600 \pi \mathrm { tt } ^ { 3 }$ 
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 
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 
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 
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)  $11.3$ 
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)  $0.7622$ 
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...

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 $x ^ { 2 } + y ^ { 2 } = 9$ .

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

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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 .

Functions and Models

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Differentiation Rules

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

Vector Calculus

Second-Order Differential Equations

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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 $x ^ { 2 } + y ^ { 2 } = 9$ .