Question 1

Evaluate the iterated integral $\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { \sqrt { \pi } } \int _ { 0 } ^ { y } e ^ { x } \sin \left( y ^ { 2 } \right) d z d y d x$

Accepted Answer

$e - 1$

Question 2

Let T be the triangle with vertices (0, 0), (2, 4), and (2, 0). Let the density at each point of T be equal to 1. Find $I _ { 0 }$ for T.

Accepted Answer

A)  $8$ 
B)  $\frac { 56 } { 3 }$ 
C)  $\frac { 16 } { 3 }$ 
D)  $\frac { 32 } { 3 }$ 
A)  $8$ 
B)  $\frac { 56 } { 3 }$ 
C)  $\frac { 16 } { 3 }$ 
D)  $\frac { 32 } { 3 }$ B

Question 3

Rewrite the following integral, switching the order of the y and z integrations. $$\int _ { 0 } ^ { 2 } \int _ { - \sqrt { 4 - x ^ { 2 } } } ^ { \sqrt { 4 - x ^ { 2 } } } \int _ { - \sqrt { 4 - x ^ { 2 } - z ^ { 2 } } } ^ { \sqrt { 4 - x ^ { 2 } - z ^ { 2 } } } f ( x , y , z ) d y d z d x$$

Accepted Answer

$\int _ { 0 } ^ { 2 } \int _ { - \sqrt { 4 - x ^ { 2 } } } ^ { \sqrt { 4 - x ^ { 2 } } } \int _ { - \sqrt { 4 - x ^ { 2 } - y ^ { 2 } } } ^ { \sqrt { 4 - x ^ { 2 } - y ^ { 2 } } } f ( x , y , z ) d z d y d x$

Question 4

Evaluate the double integral $\iint _ { R } x y + 1 d A$ where $R = \{ ( x , y ) : 1 \leq x \leq 3 \text { and } - 1 \leq \mathrm { y } \leq 2 \}$

Accepted Answer

The answer of Evaluate the double integral \(\iint _ {...

Question 5

Find the area inside the circle $x ^ { 2 } + y ^ { 2 } = 4$ and to the right of $x = 1$

Accepted Answer

The answer of Find the area inside the circle \(x...

Question 6

Evaluate the integral $\iint _ { R } x \sin \left( x ^ { 3 } \right) d A$ over the triangle with vertices (0, 0), (1, 3), and (1, -3).

Accepted Answer

A)  $2$ 
B)  $2 \cos ( 1 ) - 2$ 
C)  $0$ 
D)  $2 - 2 \cos ( 1 )$ 
A)  $2$ 
B)  $2 \cos ( 1 ) - 2$ 
C)  $0$ 
D)  $2 - 2 \cos ( 1 )$

Question 7

Find the signed volume between the graph of the given function and the specified rectangle. $$f ( x , y ) = \frac { x ^ { 2 } } { y }$$ and $R = \{ ( x , y ) : 0 \leq x \leq 2 \text { and } 1 \leq \mathrm { y } \leq e \}$

Accepted Answer

The answer of Find the signed volume between the graph...

Question 8

Use cylindrical coordinates to find $\iiint _ { R } d V$ , where $R$ is the intersection of the sphere of radius 2 and the interior of the cone $\varphi = \frac { \pi } { 4 }$

Accepted Answer

The answer of Use cylindrical coordinates to find \(\iiint _...

Question 9

Find the mass of the solid in the first octant bounded by the coordinate planes and the plane $x + y + z = 1$ , where the density is equal to the distance from the yz-plane.

Accepted Answer

The answer of Find the mass of the solid in...

Question 10

Find the volume of the solid bounded by the graph of $z = 16 - x ^ { 2 } - y ^ { 2 }$ and the xy plane.

Accepted Answer

A)  $64 \pi$ 
B)  $128 \pi$ 
C)  $192 \pi$ 
D)  $256 \pi$ 
A)  $64 \pi$ 
B)  $128 \pi$ 
C)  $192 \pi$ 
D)  $256 \pi$

Question 11

Evaluate the sum $\sum _ { i = 1 } ^ { 3 } \sum _ { j = 1 } ^ { 2 } j ^ { i }$

Accepted Answer

A)  17 
B)  6 
C)  36 
D)  20 
A)  17 
B)  6 
C)  36 
D)  20

Question 12

Evaluate the sum $\sum _ { i = 1 } ^ { 2 } \sum _ { j = 1 } ^ { 3 } j ^ { i }$

Accepted Answer

The answer of Evaluate the sum \(\sum _ { i...

Question 13

Give the spherical coordinates for the point with the rectangular coordinates $( 1,1,1 )$

Accepted Answer

The answer of Give the spherical coordinates for the point...

Question 14

Set up $\iint _ { R } f ( x , y ) d A$ as an iterated integral (or more, if necessary) where you integrate first with respect to $y$ , where $R = \{ ( x , y ) : 0 \leq x \leq 2 \text { and } 0 \leq \mathrm { y } \leq 2 x \}$

Accepted Answer

A)  $\int _ { 0 } ^ { 2 } \int _ { 2 x } ^ { 2 } f ( x , y ) d y d x$ 
B)  $\int_{-2}^{2 x^{2}} \int_{-2}^{{+2}}f(x, y) d y d x$ 
C)  $\int _ { 0 } ^ { 2 } \int _ { - \sqrt { 4 - x ^ { 2 } } } ^ { \sqrt { 4 - x ^ { 2 } } } f ( x , y ) d y d x$ 
D)  $\int _ { - 2 } ^ { 2 } \int _ { - \sqrt { 4 - x ^ { 2 } } } ^ { \sqrt { 4 - x ^ { 2 } } } f ( x , y ) d y d x$ 
A)  $\int _ { 0 } ^ { 2 } \int _ { 2 x } ^ { 2 } f ( x , y ) d y d x$ 
B)  $\int_{-2}^{2 x^{2}} \int_{-2}^{{+2}}f(x, y) d y d x$ 
C)  $\int _ { 0 } ^ { 2 } \int _ { - \sqrt { 4 - x ^ { 2 } } } ^ { \sqrt { 4 - x ^ { 2 } } } f ( x , y ) d y d x$ 
D)  $\int _ { - 2 } ^ { 2 } \int _ { - \sqrt { 4 - x ^ { 2 } } } ^ { \sqrt { 4 - x ^ { 2 } } } f ( x , y ) d y d x$

Question 15

Evaluate the iterated integral $\int _ { 3 } ^ { 4 } \int _ { 1 } ^ { 2 } \int _ { 1 } ^ { y } \frac { x } { y z } d z d y d x$

Accepted Answer

A)  $\frac { 7 } { 4 } \ln ( 2 )$ 
B)  $\frac { 7 } { 4 } ( \ln ( 2 ) ) ^ { 2 }$ 
C)  $\frac { 7 } { 2 } ( \ln ( 2 ) ) ^ { 2 }$ 
D)  $\frac { 7 } { 8 } ( \ln ( 2 ) ) ^ { 2 }$ 
A)  $\frac { 7 } { 4 } \ln ( 2 )$ 
B)  $\frac { 7 } { 4 } ( \ln ( 2 ) ) ^ { 2 }$ 
C)  $\frac { 7 } { 2 } ( \ln ( 2 ) ) ^ { 2 }$ 
D)  $\frac { 7 } { 8 } ( \ln ( 2 ) ) ^ { 2 }$

Question 16

Find the volume of the solid bounded by the graph of $z = 1 - x ^ { 2 } - y ^ { 2 }$ and the xy plane.

Accepted Answer

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

Question 17

Find the volume of the solid bounded by the given function over the specified region $\Omega$ . $f ( x , y ) = x e ^ { y }$ and $\Omega = \left\{ ( x , y ) : 0 \leq x \leq 1 \text { and } x ^ { 2 } \leq y \leq 1 \right\}$

Accepted Answer

A)  $\frac { 3 } { 2 } e$ 
B)  $\frac { 1 } { 2 }$ 
C)  $\frac { 1 } { 2 } e$ 
D)  $e - \frac { 1 } { 2 }$ 
A)  $\frac { 3 } { 2 } e$ 
B)  $\frac { 1 } { 2 }$ 
C)  $\frac { 1 } { 2 } e$ 
D)  $e - \frac { 1 } { 2 }$

Question 18

The functions $x = x ( u , v )$ and $y = y ( u , v )$ are given to determine transformations from the xy-coordinate system to a uv-coordinate system. Find the Jacobian of the transformation. $$\begin{array} { l } 
x = 2 u - 4 v \
y = 2 u + 4 v
\end{array}$$

Accepted Answer

A)  8 
B)  16 
C)  0 
D)  12 
A)  8 
B)  16 
C)  0 
D)  12

Question 19

Find the volume of the solid bounded below by the graph of $z = x ^ { 2 } + y ^ { 2 }$ and above by the plane $z = 4$

Accepted Answer

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

Question 20

Evaluate the iterated integral $\int _ { 0 } ^ { 1 } \int _ { 2 } ^ { 3 } \int _ { - 1 } ^ { 1 } x y z ^ { 2 } d z d y d x$

Accepted Answer

The answer of Evaluate the iterated integral \(\int _ {...

Evaluate the iterated integral $\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { \sqrt { \pi } } \int _ { 0 } ^ { y } e ^ { x } \sin \left( y ^ { 2 } \right) d z d y d x$

Let T be the triangle with vertices (0, 0), (2, 4), and (2, 0). Let the density at each point of T be equal to 1. Find $I _ { 0 }$ for T.

Rewrite the following integral, switching the order of the y and z integrations. $\int _ { 0 } ^ { 2 } \int _ { - \sqrt { 4 - x ^ { 2 } } } ^ { \sqrt { 4 - x ^ { 2 } } } \int _ { - \sqrt { 4 - x ^ { 2 } - z ^ { 2 } } } ^ { \sqrt { 4 - x ^ { 2 } - z ^ { 2 } } } f ( x , y , z ) d y d z d x$

Evaluate the double integral $\iint _ { R } x y + 1 d A$ where $R = \{ ( x , y ) : 1 \leq x \leq 3 \text { and } - 1 \leq \mathrm { y } \leq 2 \}$

Find the area inside the circle $x ^ { 2 } + y ^ { 2 } = 4$ and to the right of $x = 1$

Evaluate the integral $\iint _ { R } x \sin \left( x ^ { 3 } \right) d A$ over the triangle with vertices (0, 0), (1, 3), and (1, -3).

Find the signed volume between the graph of the given function and the specified rectangle. $f ( x , y ) = \frac { x ^ { 2 } } { y }$ and $R = \{ ( x , y ) : 0 \leq x \leq 2 \text { and } 1 \leq \mathrm { y } \leq e \}$

Use cylindrical coordinates to find $\iiint _ { R } d V$ , where $R$ is the intersection of the sphere of radius 2 and the interior of the cone $\varphi = \frac { \pi } { 4 }$

Find the mass of the solid in the first octant bounded by the coordinate planes and the plane $x + y + z = 1$ , where the density is equal to the distance from the yz-plane.

Find the volume of the solid bounded by the graph of $z = 16 - x ^ { 2 } - y ^ { 2 }$ and the xy plane.

Evaluate the sum $\sum _ { i = 1 } ^ { 3 } \sum _ { j = 1 } ^ { 2 } j ^ { i }$

Evaluate the sum $\sum _ { i = 1 } ^ { 2 } \sum _ { j = 1 } ^ { 3 } j ^ { i }$

Give the spherical coordinates for the point with the rectangular coordinates $( 1,1,1 )$

Set up $\iint _ { R } f ( x , y ) d A$ as an iterated integral (or more, if necessary) where you integrate first with respect to $y$ , where $R = \{ ( x , y ) : 0 \leq x \leq 2 \text { and } 0 \leq \mathrm { y } \leq 2 x \}$

Evaluate the iterated integral $\int _ { 3 } ^ { 4 } \int _ { 1 } ^ { 2 } \int _ { 1 } ^ { y } \frac { x } { y z } d z d y d x$

Find the volume of the solid bounded by the graph of $z = 1 - x ^ { 2 } - y ^ { 2 }$ and the xy plane.

Find the volume of the solid bounded by the given function over the specified region $\Omega$ . $f ( x , y ) = x e ^ { y }$ and $\Omega = \left\{ ( x , y ) : 0 \leq x \leq 1 \text { and } x ^ { 2 } \leq y \leq 1 \right\}$

The functions $x = x ( u , v )$ and $y = y ( u , v )$ are given to determine transformations from the xy-coordinate system to a uv-coordinate system. Find the Jacobian of the transformation. x=2u-4v y=2u+4v

Find the volume of the solid bounded below by the graph of $z = x ^ { 2 } + y ^ { 2 }$ and above by the plane $z = 4$

Evaluate the iterated integral $\int _ { 0 } ^ { 1 } \int _ { 2 } ^ { 3 } \int _ { - 1 } ^ { 1 } x y z ^ { 2 } d z d y d x$

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Exam 13: Double and Triple Integrals

Evaluate the iterated integral ∫01∫0π∫0yexsin⁡(y2)dzdydx\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { \sqrt { \pi } } \int _ { 0 } ^ { y } e ^ { x } \sin \left( y ^ { 2 } \right) d z d y d x∫01​∫0π​​∫0y​exsin(y2)dzdydx

Let T be the triangle with vertices (0, 0), (2, 4), and (2, 0). Let the density at each point of T be equal to 1. Find I0I _ { 0 }I0​ for T.

Evaluate the double integral ∬Rxy+1dA\iint _ { R } x y + 1 d A∬R​xy+1dA where R={(x,y):1≤x≤3 and −1≤y≤2}R = \{ ( x , y ) : 1 \leq x \leq 3 \text { and } - 1 \leq \mathrm { y } \leq 2 \}R={(x,y):1≤x≤3 and −1≤y≤2}

Find the area inside the circle x2+y2=4x ^ { 2 } + y ^ { 2 } = 4x2+y2=4 and to the right of x=1x = 1x=1

Evaluate the integral ∬Rxsin⁡(x3)dA\iint _ { R } x \sin \left( x ^ { 3 } \right) d A∬R​xsin(x3)dA over the triangle with vertices (0, 0), (1, 3), and (1, -3).

Use cylindrical coordinates to find ∭RdV\iiint _ { R } d V∭R​dV , where RRR is the intersection of the sphere of radius 2 and the interior of the cone φ=π4\varphi = \frac { \pi } { 4 }φ=4π​

Find the mass of the solid in the first octant bounded by the coordinate planes and the plane x+y+z=1x + y + z = 1x+y+z=1 , where the density is equal to the distance from the yz-plane.

Find the volume of the solid bounded by the graph of z=16−x2−y2z = 16 - x ^ { 2 } - y ^ { 2 }z=16−x2−y2 and the xy plane.

Evaluate the sum ∑i=13∑j=12ji\sum _ { i = 1 } ^ { 3 } \sum _ { j = 1 } ^ { 2 } j ^ { i }∑i=13​∑j=12​ji

Evaluate the sum ∑i=12∑j=13ji\sum _ { i = 1 } ^ { 2 } \sum _ { j = 1 } ^ { 3 } j ^ { i }∑i=12​∑j=13​ji

Give the spherical coordinates for the point with the rectangular coordinates (1,1,1)( 1,1,1 )(1,1,1)

Evaluate the iterated integral ∫34∫12∫1yxyzdzdydx\int _ { 3 } ^ { 4 } \int _ { 1 } ^ { 2 } \int _ { 1 } ^ { y } \frac { x } { y z } d z d y d x∫34​∫12​∫1y​yzx​dzdydx

Find the volume of the solid bounded by the graph of z=1−x2−y2z = 1 - x ^ { 2 } - y ^ { 2 }z=1−x2−y2 and the xy plane.

The functions x=x(u,v)x = x ( u , v )x=x(u,v) and y=y(u,v)y = y ( u , v )y=y(u,v) are given to determine transformations from the xy-coordinate system to a uv-coordinate system. Find the Jacobian of the transformation. x=2u-4v y=2u+4v

Find the volume of the solid bounded below by the graph of z=x2+y2z = x ^ { 2 } + y ^ { 2 }z=x2+y2 and above by the plane z=4z = 4z=4

Evaluate the iterated integral ∫01∫23∫−11xyz2dzdydx\int _ { 0 } ^ { 1 } \int _ { 2 } ^ { 3 } \int _ { - 1 } ^ { 1 } x y z ^ { 2 } d z d y d x∫01​∫23​∫−11​xyz2dzdydx

Limits

Derivatives

Applications of the Derivative

Definite Integrals

Techniques of Integration

Applications of Integration

Sequences and Series

Power Series

Parametric Equations Polar Coordinates and Conic Sections

Vectors

Vector Functions

Multivariable Functions

Vector Analysis

Functions and Precalculus

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Evaluate the iterated integral $\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { \sqrt { \pi } } \int _ { 0 } ^ { y } e ^ { x } \sin \left( y ^ { 2 } \right) d z d y d x$

Let T be the triangle with vertices (0, 0), (2, 4), and (2, 0). Let the density at each point of T be equal to 1. Find $I _ { 0 }$ for T.

Evaluate the double integral $\iint _ { R } x y + 1 d A$ where $R = \{ ( x , y ) : 1 \leq x \leq 3 \text { and } - 1 \leq \mathrm { y } \leq 2 \}$

Find the area inside the circle $x ^ { 2 } + y ^ { 2 } = 4$ and to the right of $x = 1$

Evaluate the integral $\iint _ { R } x \sin \left( x ^ { 3 } \right) d A$ over the triangle with vertices (0, 0), (1, 3), and (1, -3).

Use cylindrical coordinates to find $\iiint _ { R } d V$ , where $R$ is the intersection of the sphere of radius 2 and the interior of the cone $\varphi = \frac { \pi } { 4 }$

Find the mass of the solid in the first octant bounded by the coordinate planes and the plane $x + y + z = 1$ , where the density is equal to the distance from the yz-plane.

Find the volume of the solid bounded by the graph of $z = 16 - x ^ { 2 } - y ^ { 2 }$ and the xy plane.

Evaluate the sum $\sum _ { i = 1 } ^ { 3 } \sum _ { j = 1 } ^ { 2 } j ^ { i }$

Evaluate the sum $\sum _ { i = 1 } ^ { 2 } \sum _ { j = 1 } ^ { 3 } j ^ { i }$

Give the spherical coordinates for the point with the rectangular coordinates $( 1,1,1 )$

Evaluate the iterated integral $\int _ { 3 } ^ { 4 } \int _ { 1 } ^ { 2 } \int _ { 1 } ^ { y } \frac { x } { y z } d z d y d x$

Find the volume of the solid bounded by the graph of $z = 1 - x ^ { 2 } - y ^ { 2 }$ and the xy plane.

The functions $x = x ( u , v )$ and $y = y ( u , v )$ are given to determine transformations from the xy-coordinate system to a uv-coordinate system. Find the Jacobian of the transformation. x=2u-4v y=2u+4v

Find the volume of the solid bounded below by the graph of $z = x ^ { 2 } + y ^ { 2 }$ and above by the plane $z = 4$

Evaluate the iterated integral $\int _ { 0 } ^ { 1 } \int _ { 2 } ^ { 3 } \int _ { - 1 } ^ { 1 } x y z ^ { 2 } d z d y d x$