Question 1

Find the mass of the solid whose density is equal to twice the distance from the origin, which is outside the sphere of radius 3 and inside the sphere of radius 5.

Accepted Answer

A)  $544 \pi$ 
B)  $1088 \pi$ 
C)  $272 \pi$ 
D)  $136 \pi$ 
A)  $544 \pi$ 
B)  $1088 \pi$ 
C)  $272 \pi$ 
D)  $136 \pi$

Question 2

Find the mass of the solid in the first octant bounded by the coordinate planes and the plane $x + 4 y + z = 4$ , where the density is equal to the distance from the xz-plane.

Accepted Answer

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

Question 3

Give the rectangular coordinates for the point with the spherical coordinates $\left( 4 , \frac { \pi } { 6 } , \frac { \pi } { 4 } \right)$

Accepted Answer

A)  $( \sqrt { 6 } , \sqrt { 6 } , 2 \sqrt { 2 } )$ 
B)  $( \sqrt { 6 } , 2 \sqrt { 6 } , \sqrt { 2 } )$ 
C)  $( 2 \sqrt { 6 } , 2 \sqrt { 6 } , 2 \sqrt { 2 } )$ 
D)  $( \sqrt { 6 } , \sqrt { 2 } , 2 \sqrt { 2 } )$ 
A)  $( \sqrt { 6 } , \sqrt { 6 } , 2 \sqrt { 2 } )$ 
B)  $( \sqrt { 6 } , 2 \sqrt { 6 } , \sqrt { 2 } )$ 
C)  $( 2 \sqrt { 6 } , 2 \sqrt { 6 } , 2 \sqrt { 2 } )$ 
D)  $( \sqrt { 6 } , \sqrt { 2 } , 2 \sqrt { 2 } )$

Question 4

Give the iterated integral as an iterated integral or sum of iterated integrals in the opposite order of integration. $\int _ { 0 } ^ { 1 } \int _ { - x } ^ { x } f ( x , y ) d y d x$

Accepted Answer

The answer of Give the iterated integral as an iterated...

Question 5

Let D be the upper half of the unit disk. Assume it has density $\rho ( x , y ) = \sqrt { x ^ { 2 } + y ^ { 2 } }$ Find the mass of D.

Accepted Answer

The answer of Let D be the upper half of...

Question 6

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

Accepted Answer

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

Question 7

Find the (signed) volume of the solid bounded by the given function over the specified region $\Omega$ . $f ( x , y ) = x y ^ { 2 }$ and $\Omega = \left\{ ( x , y ) : x ^ { 2 } + y ^ { 2 } \leq 4 \right\}$

Accepted Answer

The answer of Find the (signed) volume of the solid...

Question 8

Find the area enclosed in one petal of $r = \sin ( 5 \theta )$

Accepted Answer

A)  $\frac { \pi } { 15 }$ 
B)  $\frac { \pi } { 20 }$ 
C)  $\frac { \pi } { 25 }$ 
D)  $\frac { \pi } { 30 }$ 
A)  $\frac { \pi } { 15 }$ 
B)  $\frac { \pi } { 20 }$ 
C)  $\frac { \pi } { 25 }$ 
D)  $\frac { \pi } { 30 }$

Question 9

Find the area enclosed in one petal of $r = \sin ( 3 \theta )$

Accepted Answer

The answer of Find the area enclosed in one petal...

Question 10

Set up the double integral $\iint _ { R } ( y - 2 x ) ^ { 3 } d A$ over the parallelogram with vertices (2, 1), (3, 3), (5, 2), and (6, 4) using the transformation $$\begin{array} { l } 
u = y - \frac { x } { 3 } \
v = y - 2 x
\end{array}$$

Accepted Answer

A)  $\int _ { \frac { 1 } { 3 } } ^ { 2 } \int _ { - 8 } ^ { - 3 } v ^ { 3 } d v d u$ 
B)  $\frac { 3 } { 5 } \int _ { \frac { 1 } { 3 } } ^ { 2 } \int _ { - 8 } ^ { - 3 } v ^ { 3 } d v d u$ 
C)  $\int _ { \frac { 1 } { 3 } } ^ { 2 } \int _ { - 8 } ^ { - 3 } u ^ { 3 } d v d u$ 
D)  $\frac { 5 } { 3 } \int _ { \frac { 1 } { 3 } } ^ { 2 } \int _ { - 8 } ^ { - 3 } v ^ { 3 } d v d u$ 
A)  $\int _ { \frac { 1 } { 3 } } ^ { 2 } \int _ { - 8 } ^ { - 3 } v ^ { 3 } d v d u$ 
B)  $\frac { 3 } { 5 } \int _ { \frac { 1 } { 3 } } ^ { 2 } \int _ { - 8 } ^ { - 3 } v ^ { 3 } d v d u$ 
C)  $\int _ { \frac { 1 } { 3 } } ^ { 2 } \int _ { - 8 } ^ { - 3 } u ^ { 3 } d v d u$ 
D)  $\frac { 5 } { 3 } \int _ { \frac { 1 } { 3 } } ^ { 2 } \int _ { - 8 } ^ { - 3 } v ^ { 3 } d v d u$

Question 11

Evaluate the double integral $\iint _ { R } \frac { x ^ { 2 } } { y } d A$ where $R = \{ ( x , y ) : 0 \leq x \leq 2 \text { and } 1 \leq \mathrm { y } \leq e \}$

Accepted Answer

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

Question 12

Give the iterated integral as an iterated integral or sum of iterated integrals in the opposite order of integration. $\int _ { - 1 } ^ { 1 } \int _ { x ^ { 2 } } ^ { 1 } f ( x , y ) d y d x$

Accepted Answer

A)  $\int _ { 0 } ^ { 1 } \int _ { - \sqrt { y } } ^ { \sqrt { y } } f ( x , y ) d x d y$ 
B)  $\int _ { - 1 } ^ { 1 } \int _ { 0 } ^ { \sqrt { y } } f ( x , y ) d x d y$ 
C)  $\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { \sqrt { y } } f ( x , y ) d x d y$ 
D)  $\int _ { - 1 } ^ { 1 } \int _ { - y ^ { 2 } } ^ { y ^ { 2 } } f ( x , y ) d x d y$ 
A)  $\int _ { 0 } ^ { 1 } \int _ { - \sqrt { y } } ^ { \sqrt { y } } f ( x , y ) d x d y$ 
B)  $\int _ { - 1 } ^ { 1 } \int _ { 0 } ^ { \sqrt { y } } f ( x , y ) d x d y$ 
C)  $\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { \sqrt { y } } f ( x , y ) d x d y$ 
D)  $\int _ { - 1 } ^ { 1 } \int _ { - y ^ { 2 } } ^ { y ^ { 2 } } f ( x , y ) d x d y$

Question 13

Give the rectangular coordinates for the point with the spherical coordinates $\left( \sqrt { 3 } , \frac { \pi } { 4 } , \arccos \left( \frac { 1 } { \sqrt { 3 } } \right) \right)$

Accepted Answer

The answer of Give the rectangular 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 $X$ , where $R = \{ ( x , y ) : 0 \leq x \leq 2 \text { and } 0 \leq \mathrm { y } \leq 2 x \}$

Accepted Answer

The answer of Set up \(\iint _ { R }...

Question 15

Write the cylindrical equation $r = 16$ in rectangular coordinates.

Accepted Answer

The answer of Write the cylindrical equation $r = 16$...

Question 16

Evaluate the following $\iiint _ { R } x y \sin ( z ) d V$ , where $R = \left\{ ( x , y , z ) : 0 \leq x \leq 1,0 \leq y \leq 2 , \text { and } 0 \leq z \leq \frac { \pi } { 2 } \right\}$

Accepted Answer

The answer of Evaluate the following \(\iiint _ { R...

Question 17

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

Accepted Answer

The answer of Set up \(\iint _ { R }...

Question 18

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

Accepted Answer

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

Question 19

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

Accepted Answer

The answer of Evaluate the integral \(\int _ { 0...

Question 20

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 the point's distance from the x-axis. Find $M _ { y }$ for T.

Accepted Answer

A)  $\frac { 32 } { 3 }$ 
B)  $16$ 
C)  $8$ 
D)  $4$ 
A)  $\frac { 32 } { 3 }$ 
B)  $16$ 
C)  $8$ 
D)  $4$

Find the mass of the solid whose density is equal to twice the distance from the origin, which is outside the sphere of radius 3 and inside the sphere of radius 5.

Find the mass of the solid in the first octant bounded by the coordinate planes and the plane $x + 4 y + z = 4$ , where the density is equal to the distance from the xz-plane.

Give the rectangular coordinates for the point with the spherical coordinates $\left( 4 , \frac { \pi } { 6 } , \frac { \pi } { 4 } \right)$

Give the iterated integral as an iterated integral or sum of iterated integrals in the opposite order of integration. $\int _ { 0 } ^ { 1 } \int _ { - x } ^ { x } f ( x , y ) d y d x$

Let D be the upper half of the unit disk. Assume it has density $\rho ( x , y ) = \sqrt { x ^ { 2 } + y ^ { 2 } }$ Find the mass of D.

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

Find the (signed) volume of the solid bounded by the given function over the specified region $\Omega$ . $f ( x , y ) = x y ^ { 2 }$ and $\Omega = \left\{ ( x , y ) : x ^ { 2 } + y ^ { 2 } \leq 4 \right\}$

Find the area enclosed in one petal of $r = \sin ( 5 \theta )$

Find the area enclosed in one petal of $r = \sin ( 3 \theta )$

Set up the double integral $\iint _ { R } ( y - 2 x ) ^ { 3 } d A$ over the parallelogram with vertices (2, 1), (3, 3), (5, 2), and (6, 4) using the transformation u=y- v=y-2x

Evaluate the double integral $\iint _ { R } \frac { x ^ { 2 } } { y } d A$ where $R = \{ ( x , y ) : 0 \leq x \leq 2 \text { and } 1 \leq \mathrm { y } \leq e \}$

Give the iterated integral as an iterated integral or sum of iterated integrals in the opposite order of integration. $\int _ { - 1 } ^ { 1 } \int _ { x ^ { 2 } } ^ { 1 } f ( x , y ) d y d x$

Give the rectangular coordinates for the point with the spherical coordinates $\left( \sqrt { 3 } , \frac { \pi } { 4 } , \arccos \left( \frac { 1 } { \sqrt { 3 } } \right) \right)$

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

Write the cylindrical equation $r = 16$ in rectangular coordinates.

Evaluate the following $\iiint _ { R } x y \sin ( z ) d V$ , where $R = \left\{ ( x , y , z ) : 0 \leq x \leq 1,0 \leq y \leq 2 , \text { and } 0 \leq z \leq \frac { \pi } { 2 } \right\}$

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

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

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

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 the point's distance from the x-axis. Find $M _ { y }$ for T.

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Exam 13: Double and Triple Integrals

Find the mass of the solid whose density is equal to twice the distance from the origin, which is outside the sphere of radius 3 and inside the sphere of radius 5.

Find the mass of the solid in the first octant bounded by the coordinate planes and the plane x+4y+z=4x + 4 y + z = 4x+4y+z=4 , where the density is equal to the distance from the xz-plane.

Give the rectangular coordinates for the point with the spherical coordinates (4,π6,π4)\left( 4 , \frac { \pi } { 6 } , \frac { \pi } { 4 } \right)(4,6π​,4π​)

Give the iterated integral as an iterated integral or sum of iterated integrals in the opposite order of integration. ∫01∫−xxf(x,y)dydx\int _ { 0 } ^ { 1 } \int _ { - x } ^ { x } f ( x , y ) d y d x∫01​∫−xx​f(x,y)dydx

Let D be the upper half of the unit disk. Assume it has density ρ(x,y)=x2+y2\rho ( x , y ) = \sqrt { x ^ { 2 } + y ^ { 2 } }ρ(x,y)=x2+y2​ Find the mass of D.

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

Find the (signed) volume of the solid bounded by the given function over the specified region Ω\OmegaΩ . f(x,y)=xy2f ( x , y ) = x y ^ { 2 }f(x,y)=xy2 and Ω={(x,y):x2+y2≤4}\Omega = \left\{ ( x , y ) : x ^ { 2 } + y ^ { 2 } \leq 4 \right\}Ω={(x,y):x2+y2≤4}

Find the area enclosed in one petal of r=sin⁡(5θ)r = \sin ( 5 \theta )r=sin(5θ)

Find the area enclosed in one petal of r=sin⁡(3θ)r = \sin ( 3 \theta )r=sin(3θ)

Set up the double integral ∬R(y−2x)3dA\iint _ { R } ( y - 2 x ) ^ { 3 } d A∬R​(y−2x)3dA over the parallelogram with vertices (2, 1), (3, 3), (5, 2), and (6, 4) using the transformation u=y- v=y-2x

Evaluate the double integral ∬Rx2ydA\iint _ { R } \frac { x ^ { 2 } } { y } d A∬R​yx2​dA where R={(x,y):0≤x≤2 and 1≤y≤e}R = \{ ( x , y ) : 0 \leq x \leq 2 \text { and } 1 \leq \mathrm { y } \leq e \}R={(x,y):0≤x≤2 and 1≤y≤e}

Give the iterated integral as an iterated integral or sum of iterated integrals in the opposite order of integration. ∫−11∫x21f(x,y)dydx\int _ { - 1 } ^ { 1 } \int _ { x ^ { 2 } } ^ { 1 } f ( x , y ) d y d x∫−11​∫x21​f(x,y)dydx

Give the rectangular coordinates for the point with the spherical coordinates (3,π4,arccos⁡(13))\left( \sqrt { 3 } , \frac { \pi } { 4 } , \arccos \left( \frac { 1 } { \sqrt { 3 } } \right) \right)(3​,4π​,arccos(3​1​))

Write the cylindrical equation r=16r = 16r=16 in rectangular coordinates.

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

Evaluate the integral ∫01∫2xx2xydydx\int _ { 0 } ^ { 1 } \int _ { 2 x } ^ { x ^ { 2 } } x y d y d x∫01​∫2xx2​xydydx

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 the point's distance from the x-axis. Find MyM _ { y }My​ for T.

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

Filters

Find the mass of the solid in the first octant bounded by the coordinate planes and the plane $x + 4 y + z = 4$ , where the density is equal to the distance from the xz-plane.

Give the rectangular coordinates for the point with the spherical coordinates $\left( 4 , \frac { \pi } { 6 } , \frac { \pi } { 4 } \right)$

Give the iterated integral as an iterated integral or sum of iterated integrals in the opposite order of integration. $\int _ { 0 } ^ { 1 } \int _ { - x } ^ { x } f ( x , y ) d y d x$

Let D be the upper half of the unit disk. Assume it has density $\rho ( x , y ) = \sqrt { x ^ { 2 } + y ^ { 2 } }$ Find the mass of D.

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

Find the (signed) volume of the solid bounded by the given function over the specified region $\Omega$ . $f ( x , y ) = x y ^ { 2 }$ and $\Omega = \left\{ ( x , y ) : x ^ { 2 } + y ^ { 2 } \leq 4 \right\}$

Find the area enclosed in one petal of $r = \sin ( 5 \theta )$

Find the area enclosed in one petal of $r = \sin ( 3 \theta )$

Set up the double integral $\iint _ { R } ( y - 2 x ) ^ { 3 } d A$ over the parallelogram with vertices (2, 1), (3, 3), (5, 2), and (6, 4) using the transformation u=y- v=y-2x

Evaluate the double integral $\iint _ { R } \frac { x ^ { 2 } } { y } d A$ where $R = \{ ( x , y ) : 0 \leq x \leq 2 \text { and } 1 \leq \mathrm { y } \leq e \}$

Give the iterated integral as an iterated integral or sum of iterated integrals in the opposite order of integration. $\int _ { - 1 } ^ { 1 } \int _ { x ^ { 2 } } ^ { 1 } f ( x , y ) d y d x$

Give the rectangular coordinates for the point with the spherical coordinates $\left( \sqrt { 3 } , \frac { \pi } { 4 } , \arccos \left( \frac { 1 } { \sqrt { 3 } } \right) \right)$

Write the cylindrical equation $r = 16$ in rectangular coordinates.

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

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

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 the point's distance from the x-axis. Find $M _ { y }$ for T.