Deck 13: Double and Triple Integrals

Question 1

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

Answer 1

18

Question 2

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

Answer 2

42

Question 3

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

Answer 3

20

Question 4

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

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

Question 5

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

Question 6

Evaluate the sum $\sum _ { i = 1 } ^ { 3 } \sum _ { j = 1 } ^ { 3 } \sum _ { k = 1 } ^ { 2 } ( k - j )$

Question 7

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

Question 8

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

Question 9

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

A) $\frac { 1 } { 8 }$
B) $\frac { 1 } { 4 }$
C) $\frac { 65 } { 2 }$
D) $\frac { 1 } { 6 }$

Question 10

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

Question 11

Find the signed volume between the graph of the given function and the specified rectangle. $$f ( x , y ) = x y + 1$$ and $R = \{ ( x , y ) : 1 \leq x \leq 3 \text { and } - 1 \leq \mathrm { y } \leq 2 \}$

Question 12

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

Question 13

Find the signed volume between the graph of the given function and the specified rectangle. $$f ( x , y ) = x ^ { 2 } y ^ { 3 }$$ and $R = \{ ( x , y ) : 1 \leq x \leq 3 \text { and } - 1 \leq \mathrm { y } \leq 2 \}$

A) $\frac { 1 } { 8 }$
B) $\frac { 1 } { 4 }$
C) $\frac { 65 } { 2 }$
D) $\frac { 1 } { 6 }$

Question 14

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

Question 15

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

A) $0$
B) $\frac { 1 } { 3 }$
C) $\frac { 2 } { 3 }$
D) $1$

Question 16

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 $R = \{ ( x , y ) : 0 \leq x \leq 2 \text { and } 0 \leq \mathrm { y } \leq 2 x \}$

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 $R = \{ ( x , y ) : 0 \leq x \leq 2 \text { and } 0 \leq \mathrm { y } \leq 2 x \}$

Question 18

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 $R = \{ ( x , y ) : 0 \leq x \leq 2 \text { and } 0 \leq \mathrm { y } \leq 2 x \}$

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 19

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 $R = \{ ( x , y ) : 0 \leq x \leq 2 \text { and } 0 \leq \mathrm { y } \leq 2 x \}$

Question 20

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

Question 21

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

Question 22

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

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 23

Find the volume of the solid bounded by the given function over the specified region $\Omega$ . $f ( x , y ) = 3 x - y + 5$ and $\Omega = \{ ( x , y ) : 0 \leq x \leq 2 \text { and } 0 \leq \mathrm { y } \leq 2 x \}$

Question 24

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

Question 25

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

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

Question 26

Find the volume of the solid bounded above by the plane $z = x + y$ and below by the triangle with vertices (0, 0, 0), (1, 1, 0), and (1, -1, 0).

Question 27

Find the volume of the solid bounded above by the plane $z = 10 - x - 2 y$ and below by the triangle with vertices (0, 0), (1, 0), and (0, 1) in the first quadrant of the xy plane.

A) $\frac { 20 } { 3 }$
B) $\frac { 23 } { 6 }$
C) $4$
D) $\frac { 8 } { 3 }$

Question 28

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

Question 29

Evaluate the integral $\int _ { 0 } ^ { \frac { \pi } { 2 } } \int _ { - \sin ( x ) } ^ { \sin ( x ) } \cos ( \cos ( x ) ) d y d x$

Question 30

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).

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

Question 31

Find the area enclosed by the spiral $r = \theta$ and the y-axis for $\frac { \pi } { 2 } \leq \theta \leq \frac { 3 \pi } { 2 }$

Question 32

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

Question 33

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

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

Question 34

Find the area between the cardioids $r = 3 + 3 \sin ( \theta )$ and $r = 2 + 2 \sin ( \theta )$

Question 35

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

Question 36

Find the area between the cardioids $r = 5 + 5 \sin ( \theta )$ and $r = 2 + 2 \sin ( \theta )$

A) $\frac { 57 \pi } { 2 }$
B) $\frac { 59 \pi } { 2 }$
C) $\frac { 61 \pi } { 2 }$
D) $\frac { 63 \pi } { 2 }$

Question 37

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

Question 38

Find the volume of the solid bounded by the graph of $z = \sqrt { 4 - x ^ { 2 } - y ^ { 2 } }$ and the xy plane.

Question 39

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

Question 40

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

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

Question 41

Let T be the triangle with vertices (0, 0), (2, 4), and (2, 0). Find the centroid of T.

Question 42

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 the mass of T.

Question 43

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 $M _ { x }$ for T.

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

Question 44

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.

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

Question 45

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 $I _ { x }$ for T.

Question 46

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 $I _ { y }$ for T.

Question 47

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.

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

Question 48

A disk of radius 2 meters is covered with mites. At the edge of the disk their density is 10,000 mites per square meter and at the center the density is 20,000 mites per square meter. If the mite density varies linearly with the distance from the center, how many mites are in the disk?

Question 49

Let $\rho ( r , \theta ) = \frac { \theta } { \pi ^ { 2 } }$ be a joint probability distribution function on the unit disk. What is the probability of an event occurring in the region bounded by the spiral $r = \frac { \theta } { 4 } , 0 \leq \theta \leq \pi$ and the x-axis?

Question 50

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.

Question 51

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

Question 52

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

Question 53

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

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 54

Evaluate the following $\iiint _ { R } x y z ^ { 2 } d V$ , where $R = \{ ( x , y , z ) : 0 \leq x \leq 1,0 \leq y \leq 3 , \text { and } - 1 \leq z \leq 1 \}$

Question 55

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

Question 56

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

A) $\frac { 1 } { 2 }$
B) $\frac { 3 } { 4 }$
C) $\frac { 2 } { 3 }$
D) $\frac { 1 } { 8 }$

Question 57

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

Question 58

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

Question 59

Evaluate $\int _ { 0 } ^ { 2 } \int _ { 0 } ^ { 1 + x ^ { 2 } } \int _ { 1 - x + y } ^ { 1 + y } 2 z x d z d y d x$

Question 60

Evaluate $\int _ { 3 } ^ { 17 } \int _ { - 2 - x } ^ { 2 + x ^ { 2 } } \int _ { 4 x } ^ { x ^ { 2 } } 3 x y z ^ { 3 } d z d y d x$

A) $\frac { 34 } { 3 }$
B) $- \frac { 17 } { 42 }$
C) $0$
D) $6$

Question 61

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.

Question 62

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.

Question 63

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 $\rho ( x , y , z ) = 12 x y$

A) $\frac { 1 } { 8 }$
B) $\frac { 1 } { 10 }$
C) $\frac { 1 } { 12 }$
D) $\frac { 1 } { 14 }$

Question 64

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

Question 65

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

Question 66

Give the rectangular coordinates for the point with the cylindrical coordinates $\left( \sqrt { 2 } , \frac { \pi } { 4 } , 1 \right)$

A) $( 1,1,1 )$
B) $( 1,2,1 )$
C) $\left( 1 , \frac { 1 } { 2 } , 1 \right)$
D) $( 2,2,2 )$ d

Question 67

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

Question 68

Write $y = 2$ in cylindrical coordinates.

Question 69

Write $y = 2$ in spherical coordinates.

A) $\rho = 2 \csc ( \theta ) \sec ( \varphi )$
B) $\rho = 2 \csc ( \theta ) \csc ( \varphi )$
C) $\rho = 2 \sec ( \theta ) \csc ( \varphi )$
D) $\rho = 2 \sec ( \theta ) \sec ( \varphi )$

Question 70

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

Question 71

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

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 72

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

Question 73

Change $x ^ { 2 } + y ^ { 2 } = z$ into cylindrical coordinates.

Question 74

Use cylindrical coordinates to find $\iiint _ { R } z d V$ , where $R = \left\{ ( x , y , z ) : x ^ { 2 } + y ^ { 2 } \leq 4 , \text { and } 0 \leq \mathrm { z } \leq x ^ { 2 } + y ^ { 2 } \right\}$

Question 75

Use cylindrical coordinates to find $\iiint _ { R } d V$ , where $R = \left\{ ( r , \theta , z ) : 0 \leq \theta \leq \pi , 0 \leq \mathrm { r } \leq 4 \sin ( \theta ) , \text { and } 0 \leq \mathrm { z } \leq \sqrt { 16 - r ^ { 2 } } \right\}$

Question 76

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 $\varphi = \frac { \pi } { 4 }$

Question 77

Find the mass of the solid whose density is equal to twice the distance from the origin, which is below the plane $z = 4$ and above the cone $z ^ { 2 } = x ^ { 2 } + y ^ { 2 }$ and the xy plane.

A) $\frac { 8 } { 3 } \pi$
B) $\frac { 16 } { 3 } \pi$
C) $\frac { 32 } { 3 } \pi$
D) $\frac { 256 } { 3 } \pi ( 2 \sqrt { 2 } - 1 )$

Question 78

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.

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

Question 79

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. $$\begin{array} { l } x = 2 u - v \\y = 2 u + v\end{array}$$

Question 80

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. $$\begin{array} { l } x = 3 u + 2 v \\y = u + v\end{array}$$