Deck 12: Multiple Integrals

Question 1

Find the Jacobian of the transformation , .

Answer 1

4v

Question 2

Find the Jacobian of the transformation $x = \rho \sin \phi \cos \theta$ , $y = \rho \sin \phi \sin \theta$ , $z = \rho \cos \phi$ .

A) $\theta$
B) $\beta ^ { 2 } \sin \phi$
C) $\rho \sin \phi$
D) $\rho ^ { 2 } \cos \phi$
E)2 $\theta$
F) $\rho ^ { 2 } \sin \theta$
G) $\rho \sin \theta$
H) $\rho \cos \phi$

Answer 2

$\beta ^ { 2 } \sin \phi$

Question 3

Find the Jacobian of the transformation , .

Answer 3

-2s

Question 4

Evaluate the integral $\iint _ { R } \sqrt { x ^ { 2 } + 9 y ^ { 2 } } d A$ , where R is the region enclosed by the ellipse $\frac { x ^ { 2 } } { 9 } + y ^ { 2 } = 1$ .

A)24 $\pi$
B) $\pi$
C)12 $\pi$
D)3 $\pi$
E)10 $\pi$
F)2 $\pi$
G)6 $\pi$
H)4 $\pi$

Question 5

Find the Jacobian of the transformation $x = u ^ { 2 }$ , $y = v ^ { 3 }$ .

A) $u v ^ { 2 }$

B) $2 u v ^ { 2 }$
C) $3 u v ^ { 2 }$
D) $6 u v ^ { 2 }$
E) $- u v ^ { 2 }$
F) $- 2 u v ^ { 2 }$

G) $- 3 u v ^ { 2 }$

H) $- 6 u v ^ { 2 }$

Question 6

Find the Jacobian of the transformation x = 3u + v, y = u - 2w, z = v + w.

A)1
E)-1
B)6
F)-6
C)-5
G)5
D)4
H)-4

Question 7

Find the area of the region whose image under the transformation x = u + v, y = v - 2u is $D = \left\{ ( x , y ) \mid - 1 \leq x \leq 1,0 \leq y \leq 1 - x ^ { 2 } \right\}$ .

A) $\frac { 1 } { 9 }$

B) $\frac { 2 } { 9 }$
C) $\frac { 1 } { 3 }$
D) $\frac { 4 } { 9 }$
E) $\frac { 5 } { 9 }$
F) $\frac { 2 } { 3 }$

G) $\frac { 7 } { 9 }$

H) $\frac { 8 } { 9 }$

Question 8

Evaluate the integral $\iint _ { R } \frac { x + y } { x - y } d A$ , where R is the triangular region with vertices (1, 0), (0, -1), and (0, 0).

A)0
B) $\frac { 1 } { 4 }$
C) $\frac { 1 } { 2 }$
D) $\frac { 3 } { 4 }$
E)1
F) $\frac { 3 } { 2 }$
G)2
H)-2

Question 9

Find the Jacobian of the transformation $x = u ^ { 2 }$ , $y = v ^ { 3 }$ , when $u = \frac { 1 } { 2 }$ and v = 1.

A)1
B)2
C)3
D)6
E)-1
F)-2
G)-3
H)-6

Question 10

Find the Jacobian of the transformation x = u + v, y = 2u - v.

A)1
E)-1
B)2
F)-2
C)3
G)-3
D)4
H)-4

Question 11

Use the change of variables u = 2x - y, v = x + y to evaluate where R is the region bounded by 2x - y = 1, 2x - y = 3, x + y = 1, and x + y = 2.

Question 12

Find the Jacobian of the transformation x = u, y = 2v, z = 3w.

A)1
E)-1
B)2
F)-2
C)3
G)-3
D)6
H)-6

Question 13

Use the change of variables , to evaluate , where R is the region bounded by the curves xy = 1, xy = 2, , and .

Question 14

Under the transformation x = u + v, y = v - 2u, the image of the circle $x ^ { 2 } + y ^ { 2 } \leq 1$ is an ellipse. What is the area of that ellipse?

A) $\frac { \pi } { 4 }$

B) $\frac { \pi } { 3 }$
C) $\frac { \pi } { 2 }$
D) $\frac { 2 \pi } { 3 }$
E) $\pi$
F) $\frac { 3 \pi } { 2 }$

G) $2 \pi$

H) $3 \pi$

Question 15

Use the change of variables x = 2u + 3v, y = 3u - 2v to evaluate , where R is the square with vertices (0, 0), (2, 3), (5, 1), and (3, -2).

Question 16

Find the Jacobian of the transformation x = u sin v, y = u cos v when u = 3 and v = 5.

A)3
E)-3
B)5
F)-5
C)7.5
G)-7.5
D)15
H)-15

Question 17

Find the Jacobian of the transformation $x = r \cos \theta$ , $y = r \sin \theta$ , z = z.

A) $\theta$
B)z
C)r
D)1
E)2 $\theta$
F)2z
G)2r
H)2

Question 18

Find the Jacobian of the transformation x = 2u, , .

Question 19

Find the Jacobian of the transformation $x = u \sin v$ , $y = u \cos v$ .

A) $- u$

B)
$- u ^ { 2 }$
C)u
D) $u ^ { 2 }$
E) $- u ^ { 2 } \sin v \cos v$
F) $u ^ { 2 } \sin v \cos v$
G) $- 2 u ^ { 2 } \sin v \cos v$
H) $2 u ^ { 2 } \sin v \cos v$

Question 20

Use the change of variables , to evaluate , where R is the region bounded by the ellipse .

Question 21

Evaluate ,where R is the rectangular region bounded by the lines x + y = 0, , x - y = 0, and .

Question 22

Evaluate the iterated integral $\int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { \sqrt { 1 - r ^ { 2 } } } z r d z d r d \theta$ .

A) $\frac { \pi } { 8 }$

B) $\frac { \pi } { 4 }$

C) $\pi$

D)2 $\pi$

E) $\frac { \pi } { 16 }$

F) $\frac { \pi } { 2 }$

G) $\frac { \pi } { 3 }$

H) $\frac { \pi } { 6 }$

Question 23

Find a transformation x = x (u, v), y = y (u, v) maps the region in the uv-plane into the xy-plane.

Question 24

Let T be the transformation given by x = 2u + v, y = u + 2v.(a) A region S in the uv-plane is given below. Sketch the image R of S in the xy-plane. (b) Find the inverse transformation .(c) Evaluate the double integral .

Question 25

Evaluate , where R is the region enclosed by the ellipse .

Question 26

Compute the Jacobian of the transformation T given by , , and find the image of under T.

Question 27

Let T be the transformation given by x = 2u + v, y = 3u.(a) A region S in the uv-plane is given below. Sketch the image R of S in the xy-plane. (b) Find the inverse transformation .(c) Evaluate the double integral .

Question 28

Compute the Jacobian of the transformation T given by , . Compute the area of the image of and compare it to the area of S.

Question 29

Evaluate , where E is the solid bounded by the ellipsoid .

Question 30

Describe the image R of the set under the transformation , , and then compute .

Question 31

Find a transformation x = x (u, v), y = y (u, v) maps the region in the uv-plane into the xy-plane.

Question 32

Evaluate the iterated integral $\int _ { 0 } ^ { x / 2 } \int _ { 0 } ^ { x / 2 } \int _ { 0 } ^ { 1 } \rho ^ { 3 } \sin \phi \cos \phi d \rho d \phi d \theta$ .

A) $\frac { \pi } { 8 }$

B) $\frac { \pi } { 4 }$

C) $\pi$

D)2 $\pi$

E) $\frac { \pi } { 16 }$

F) $\frac { \pi } { 2 }$

G) $\frac { \pi } { 3 }$

H) $\frac { \pi } { 6 }$

Question 33

Evaluate the iterated integral $\int _ { 0 } ^ { 2 x } \int _ { 0 } ^ { 1 } \int _ { 1 - y ^ { 2 } } ^ { 4 } r ^ { 2 } d z d r d \theta$ .

A) $\frac { \pi } { 5 }$

B) $\frac { 7 \pi } { 5 }$

C)4 $\pi$

D)12 $\pi$

E) $\frac { \pi } { 16 }$

F) $\frac { \pi } { 2 }$

G) $\frac { 12 \pi } { 3 }$

H) $\frac { 12 \pi } { 5 }$

Question 34

Use the change of variables , , to evaluate , where E is the solid enclosed by the ellipsoid .

Question 35

Evaluate ,where R is the rectangular region bounded by the lines x + y = 0, x + y = 1, x - y = 0, and x - y = 1.

Question 36

Find a transformation x = x (u, v), y = y (u, v) maps the region in the uv-plane into the xy-plane.

Question 37

Evaluate the triple integral $\iiint _ { E } r d V$ in cylindrical coordinates, where $E = \{ ( r , \theta , z ) \mid 0 \leq r \leq 1,0 \leq \theta \leq 2 \pi , 0 \leq z \leq 1 \}$ .

A) $\frac { \pi } { 4 }$

B) $\frac { \pi } { 3 }$

C) $\frac { \pi } { 2 }$

D) $\frac { 2 \pi } { 3 }$

E) $\frac { 3 \pi } { 4 }$

F) $\pi$

G) $\frac { 4 \pi } { 3 }$

H)2 $\pi$

Question 38

Evaluate the triple integral $\iiint _ { E } 1 d V$ in cylindrical coordinates, where $E = \{ ( r , \theta , z ) \mid 0 \leq r \leq 1,0 \leq \theta \leq \pi , 0 \leq z \leq 1 \}$ .

A) $\frac { \pi } { 4 }$

B) $\frac { \pi } { 3 }$

C) $\frac { \pi } { 2 }$

D) $\frac { 2 \pi } { 3 }$

E) $\frac { 3 \pi } { 4 }$

F) $\pi$

G) $\frac { 4 \pi } { 3 }$

H)2 $\pi$

Question 39

Evaluate by making an appropriate change of variables, where R is the region in the first quadrant bounded by the ellipse .

Question 40

Compute the Jacobian of the transformation T given by $x = v \cos 2 \pi u$ , $y = v \sin 2 \pi u$ . Describe the image of $S = \{ ( u , v ) \mid 0 \leq u \leq 1,0 \leq v \leq 1 \}$ , and compute its area.

Question 41

A sphere of radius k has a volume of . Set up the iterated integrals in rectangular, cylindrical, and spherical coordinates to show this.

Question 42

Evaluate the triple integral $\iiint _ { E } \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } d V$ in spherical coordinates, where E is the solid bounded by the hemisphere $z = \sqrt { 4 - x ^ { 2 } - y ^ { 2 } }$ and the plane z = 0.

A) $\frac { \pi } { 4 }$

B)16 $\pi$

C) $\frac { \pi } { 2 }$

D) $\frac { 2 \pi } { 3 }$

E) $\frac { 3 \pi } { 4 }$

F)8 $\pi$

G) $\frac { 4 \pi } { 3 }$

H)2 $\pi$

Question 43

Evaluate , where E is the solid bounded by the sphere .

Question 44

Find the volume bounded above by the surface , , below by the xy-plane, and laterally by the cylinder .

Question 45

Use cylindrical coordinates to find , where R is the region bounded by and .

Question 46

Find the mass of the solid that occupies the region bounded by $x ^ { 2 } + y ^ { 2 } = 1$ , z = 2, and z = 0 and has density function $\rho ( x , y , z ) = z$ .

A) $\frac { \pi } { 4 }$

B) $\frac { \pi } { 3 }$

C) $\frac { \pi } { 2 }$

D) $\frac { 2 \pi } { 3 }$

E) $\frac { 3 \pi } { 4 }$

F) $\pi$

G) $\frac { 4 \pi } { 3 }$

H)2 $\pi$

Question 47

Evaluate by changing to cylindrical coordinates.

Question 48

Evaluate the triple integral $\iiint _ { E } \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) d V$ , where $E = \left\{ ( x , y , z ) \mid x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \leq 1 \right\}$ .

A) $\frac { 2 \pi } { 3 }$

B) $\frac { 2 \pi } { 5 }$

C) $\frac { 4 \pi } { 3 }$

D) $\frac { 4 \pi } { 5 }$

E)2 $\pi$

F) $\frac { 6 \pi } { 5 }$

G) $\frac { 8 \pi } { 3 }$

H) $\frac { 8 \pi } { 5 }$

Question 49

Evaluate by changing to spherical coordinates.

Question 50

Find the volume of the region above the paraboloid and below the hemisphere .

Question 51

Evaluate the iterated integral $\int _ { 0 } ^ { \pi / 2 } \int _ { 0 } ^ { \pi / 2 } \int _ { 0 } ^ { 1 } \rho ^ { 2 } \sin \phi d \rho d \phi d \theta$ .

A) $\frac { \pi } { 6 }$

B) $\frac { \pi } { 4 }$

C)4 $\pi$

D)2 $\pi$

E) $\frac { \pi } { 16 }$

F) $\frac { \pi } { 2 }$

G) $\frac { \pi } { 3 }$

H) $\frac { 4 \pi } { 3 }$

Question 52

Evaluate the triple integral $\iiint _ { E } \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) d V$ in spherical coordinates, where E is the solid in the first octant bounded by the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ and the coordinate planes.

A) $\frac { 32 \pi } { 5 }$

B) $\frac { 4 \pi } { 5 }$

C) $\frac { \pi } { 2 }$

D) $\frac { 2 \pi } { 3 }$

E) $\frac { 16 \pi } { 5 }$

F) $\pi$

G) $\frac { 4 \pi } { 3 }$

H)2 $\pi$

Question 53

Find the mass of the solid that occupies the region bounded by the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ and the plane z = 1 and has density function $\rho ( x , y , z ) = \sqrt { x ^ { 2 } + y ^ { 2 } }$ .

A) $\frac { \pi } { 4 }$

B) $\frac { \pi } { 3 }$

C) $\frac { \pi } { 2 }$

D) $\frac { 2 \pi } { 3 }$

E) $\frac { 3 \pi } { 4 }$

F) $\pi$

G) $\frac { 4 \pi } { 15 }$

H)2 $\pi$

Question 54

Find the mass of that portion of the solid bounded above by the sphere which lies in the first octant, if the density varies as the distance from the center of the sphere.

Question 55

A region W in is described completely by , , , and .(a) Describe or sketch this region. (b) Write an integral in rectangular coordinates which gives the volume of W. Do not work out this integral.(c) Write an integral in spherical coordinates which gives the volume of W. Find the volume of W using this integral.

Question 56

Evaluate by changing to spherical coordinates.

Question 57

Evaluate , where E is the solid bounded by the sphere and the cone .

Question 58

Use a triple integral in spherical coordinates to find the volume of that part of the sphere which lies inside the cone .

Question 59

Let E be the solid that lies below the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = a ^ { 2 }$ and above the cone $\phi = \beta$ , where $0 < \beta < \frac { \pi } { 2 }$ . Find the value of the triple integral $\iiint _ { E } z d V$ .

A) $\pi a ^ { 2 } \sin \beta$

B). $\frac { 1 } { 2 } \pi a ^ { 2 } \sin \beta$
C) $\pi a ^ { 2 } \sin ^ { 2 } \beta$
D) $\frac { 1 } { 2 } \pi a ^ { 2 } \sin ^ { 2 } \beta$
E) $\frac { 1 } { 4 } \pi a ^ { 2 } \sin ^ { 2 } \beta$
F) $\frac { 1 } { 2 } \pi \alpha ^ { 4 } \sin ^ { 2 } \beta$


G) $\frac { 1 } { 4 } \pi a ^ { 4 } \sin ^ { 2 } \beta$

H) $\pi a ^ { 4 } \sin \beta$

Question 60

Find the volume of the region inside the cylinder which is bounded below by the xy-plane and above by the sphere .

Question 61

Evaluate the iterated integral $\int _ { 0 } ^ { \mathrm { In } 4 } \int _ { 0 } ^ { \mathrm { In } 3 } \int _ { 0 } ^ { \mathrm { In } 2 } e ^ { 05 x + y - z } d z d y d x$ .

A)6
B)1
C)3
D)2
E)-6
F)-1
G)-3
H)-2

Question 62

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

A) $\frac { 1 } { 120 }$

B) $\frac { 1 } { 90 }$
C) $\frac { 1 } { 60 }$
D) $\frac { 1 } { 48 }$
E) $\frac { 1 } { 36 }$
F) $\frac { 1 } { 24 }$

G) $\frac { 1 } { 12 }$

H) $\frac { 1 } { 6 }$

Question 63

Give a geometric description of the solid S whose volume in cylindrical coordinates is given by .

Question 64

Find the mass of the solid that occupies the region $E = \{ ( x , y , z ) \mid 0 \leq x \leq 1,0 \leq y \leq 1,0 \leq z \leq 1 \}$ and has density function $\rho ( x , y , z ) = x$ .

A) $\frac { 1 } { 6 }$

B) $\frac { 1 } { 4 }$
C) $\frac { 1 } { 3 }$
D) $\frac { 1 } { 2 }$
E) $\frac { 2 } { 3 }$
F) $\frac { 3 } { 4 }$

G) $\frac { 5 } { 6 }$

H)1

Question 65

Find the mass of a solid ball of radius 2 if the density at each point (x, y, z) is .

Question 66

Evaluate the iterated integral $\int _ { 0 } ^ { 2 } \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { x } 6 x y ^ { 2 } \sin z d z d y d x$ .

A) $\sqrt { 2 }$
B)2
C)4
D)16
E)12
F) $2 \sqrt { 2 }$
G)48
H)8

Question 67

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

A) $\frac { 1 } { 120 }$

B) $\frac { 1 } { 90 }$
C) $\frac { 1 } { 60 }$
D) $\frac { 1 } { 48 }$
E) $\frac { 1 } { 36 }$
F) $\frac { 1 } { 24 }$

G) $\frac { 1 } { 12 }$

H) $\frac { 1 } { 6 }$

Question 68

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

A) $- \frac { 5 } { 12 }$

B) $- \frac { 5 } { 6 }$
C) $- \frac { 1 } { 12 }$
D) $- \frac { 1 } { 6 }$
E)
$\frac { 5 } { 12 }$
F) $\frac { 5 } { 6 }$

G) $\frac { 1 } { 12 }$

H) $\frac { 1 } { 6 }$

Question 69

Let E be the part of the solid ellipsoid that lies in the first octant above
the plane z = 1.(a) Express the triple integral as an iterated integral in rectangular coordinates.(b) Express the triple integral as an iterated integral in cylindrical coordinates.(c) Express the triple integral as an iterated integral in spherical coordinates.

Question 70

Evaluate the triple integral $\iiint _ { E } x d V$ , where $E = \{ ( x , y , z ) \mid 0 \leq x \leq y , 0 \leq y \leq 1,0 \leq z \leq 1 \}$ .

A) $\frac { 1 } { 6 }$

B) $\frac { 1 } { 4 }$
C) $\frac { 1 } { 3 }$
D) $\frac { 1 } { 2 }$
E) $\frac { 2 } { 3 }$
F) $\frac { 1 } { 12 }$

G) $\frac { 5 } { 6 }$

H)1

Question 71

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

A) $\frac { 1 } { 120 }$

B) $\frac { 1 } { 90 }$
C) $\frac { 1 } { 60 }$
D) $\frac { 1 } { 48 }$
E) $\frac { 1 } { 36 }$
F) $\frac { 1 } { 24 }$
G) $\frac { 1 } { 12 }$

H) $\frac { 1 } { 6 }$

Question 72

Sketch the region E whose volume is given by the integral .

Question 73

Evaluate the triple integral $\iiint _ { E } y d V$ , where E is the solid bounded by the coordinate planes and the plane 2x + y + z = 4.

A)1
B) $\frac { 3 } { 16 }$
C) $\frac { 8 } { 3 }$
D) $\frac { 3 } { 8 }$
E) $\frac { 2 } { 3 }$
F) $\frac { 16 } { 3 }$
G) $\frac { 32 } { 3 }$
H) $\frac { 3 } { 32 }$

Question 74

Evaluate , where E is the solid bounded by the cylinder , above by z = 3 and below by z = 0.

Question 75

Let E be the solid bounded above by the sphere and below by the cone .(a) Express the volume of E as an iterated integral in rectangular coordinates.(b) Express the volume of E as an iterated integral in cylindrical coordinates.(c) Express the volume of E as an iterated integral in spherical coordinates.

Question 76

Evaluate the iterated integral $\int _ { 0 } ^ { x / 4 } \int _ { 0 } ^ { 2 } \int _ { 0 } ^ { x ^ { 2 } } x \cos ( 2 y ) d z d x d y$ .

A)0
B)8
C)4
D)16
E)6
F)2
G)5
H)1

Question 77

Evaluate the triple integral $\iiint _ { E } z d V$ , where E is the wedge in the first octant bounded by $y ^ { 2 } + z ^ { 2 } = 1$ , y = x, and the yz-plane.

A) $\frac { 1 } { 4 }$

B) $\frac { 1 } { 8 }$
C) $\frac { 1 } { 2 }$
D) $\frac { 2 } { 3 }$
E) $\frac { 3 } { 4 }$
F)1
G) $\frac { 5 } { 4 }$

H) $\frac { 4 } { 3 }$

Question 78

Evaluate the triple integral $\iiint _ { E } ( x + 2 y ) d V$ , where $E = \{ ( x , y , z ) \mid 0 \leq x \leq 1,0 \leq y \leq 1,0 \leq z \leq 1 \}$ .

A) $\frac { 1 } { 4 }$

B) $\frac { 1 } { 2 }$
C) $\frac { 3 } { 4 }$
D)1
E) $\frac { 5 } { 4 }$
F) $\frac { 3 } { 2 }$

G) $\frac { 7 } { 4 }$
H)2

Question 79

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

A) $\frac { 11 } { 3 }$
B)8

C)4
D)16
E) $\frac { 11 } { 6 }$
F)2
G)5
H)1

Question 80

Give a geometric description of the solid S whose volume in spherical coordinates is given by .