Question 1

Find the volume of the indicated region.
-the region bounded below by the $x y$-plane, laterally by the cylinder $r = 4 \cos \theta$, and above by the plane $z = 3$

Accepted Answer

A)  $12 \pi$ 
B)  $3 \pi$ 
C)  $36 \pi$ 
D)  $9 \pi$

Question 2

Find the volume of the indicated region.
-$$\text { the tetrahedron bounded by the coordinate planes and the plane } \frac { x } { 9 } + \frac { y } { 8 } + \frac { z } { 10 } = 1$$

Accepted Answer

A)  240 
B)  360 
C)  180 
D)  120

Question 3

Use the given transformation to evaluate the integral.
-$$\begin{array} { l } 
x = 6 u , y = 9 \mathrm { v } , z = 2 w ; \
\quad \iiint \left( \frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 81 } + \frac { z ^ { 2 } } { 4 } \right) ^ { 3 } d x d y
\end{array}$$
where $R$ is the interior of the ellipsoid $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 81 } + \frac { z ^ { 2 } } { 4 } = 1$

Accepted Answer

A)  $36 \pi$ 
B)  $81 \pi$ 
C)  $72 \pi$ 
D)  $27 \pi$

Question 4

Evaluate the integral by changing the order of integration in an appropriate way.
-$\int _ { 0 } ^ { 10 } \int _ { 1 } ^ { 2 } \int _ { \sqrt { x / 10 } } ^ { 1 } \frac { e ^ { y ^ { 3 } } } { z } d y d z d x$

Accepted Answer

A)  $5 \ln 2 ( \mathrm { e } - 1 )$ 
B)  $\frac { 10 } { 3 } \ln 2 ( e - 1 )$ 
C)  $\frac { 10 } { 3 } \ln 3 ( \mathrm { e } - 1 )$ 
D)  $5 \ln 3 ( e - 1 )$

Question 5

Evaluate the integral.
-$$\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { 1 } ( 9 x - 8 y ) d y d x$$

Accepted Answer

A)  $\frac { 17 } { 2 }$ 
B)  $\frac { 145 } { 2 }$ 
C)  $- \frac { 17 } { 2 }$ 
D)  $\frac { 1 } { 2 }$

Question 6

$$\text { Set up the iterated integral for evaluating } \iint _ { D } \int f ( r , \theta , z ) d z r d r d \theta \text { over the given region D. }$$
-$\mathrm { D }$ is the right circular cylinder whose base is the circle $\mathrm { r } = 6 \cos \theta$ in the $x y - p l a n e$ and whose top lies in the plane $z = 8 - x - y$.

Accepted Answer

A)  $\int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { 6 \sin \theta } \int _ { 0 } ^ { 8 - \sin \theta - \cos \theta } f ( r , \theta , z ) d z r d r d \theta$ 
B)  $\int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { 6 \cos \theta } \int _ { 0 } ^ { 8 - r ( \cos \theta + \sin \theta ) } \mathrm { f } ( \mathrm { r } , \theta , \mathrm { z } ) \mathrm { dz } \mathrm { rdr } \mathrm { d } \theta$ 
C)  $\int _ { 0 } ^ { \pi } \int _ { 0 } ^ { 6 \cos \theta } \int _ { 0 } ^ { 8 - r ( \cos \theta + \sin \theta ) } f ( r , \theta , z ) d z r d r d \theta$ 
D)  $\int _ { 0 } ^ { \pi } \int _ { 0 } ^ { 6 \sin \theta } \int _ { 0 } ^ { 8 - \sin \theta - \cos \theta } \mathrm { f } ( \mathrm { r } , \theta , \mathrm { z } ) \mathrm { dz } \mathrm { rdr } \mathrm { d } \theta$

Question 7

Evaluate the improper integral.
-$$\int _ { 0 } ^ { \infty } \int _ { 0 } ^ { \infty } \frac { d x d y } { ( x + 3 ) ^ { 2 } ( y + 10 ) ^ { 2 } }$$

Accepted Answer

A)  $\frac { 1 } { 30 }$ 
B)  $\frac { 1 } { 60 }$ 
C)  $\frac { 1 } { 120 }$ 
D)  $\frac { 1 } { 90 }$

Question 8

Change the Cartesian integral to an equivalent polar integral, and then evaluate.
-$\int _ { - 3 } ^ { 3 } \int _ { - \sqrt { 9 - x ^ { 2 } } } ^ { \sqrt { 9 - x ^ { 2 } } } \frac { 1 } { \left( 1 + x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } } d y d x$

Accepted Answer

A)  $\frac { 18 } { 5 } \pi$ 
B)  $\frac { 9 } { 5 } \pi$ 
C)  $\frac { 9 } { 19 } \pi$ 
D)  $\frac { 9 } { 10 } \pi$

Question 9

Evaluate the integral.
-$$\int _ { 0 } ^ { \ln 6 } \int _ { e ^ { y } } ^ { 6 } e ^ { y } d x d y$$

Accepted Answer

A)  $\frac { 49 } { 4 }$ 
B)  $\frac { 25 } { 4 }$ 
C)  $\frac { 49 } { 2 }$ 
D)  $\frac { 25 } { 2 }$

Question 10

Find the average value of F(x, y, z) over the given region.
-F(x, y, z) = 9x + 10y + 3z over the rectangular solid in the first octant bounded by the coordinate planes and the planes x = 4, y = 7, z = 6

Accepted Answer

A)  97 
B)  $\frac { 124 } { 3 }$ 
C)  62 
D)  132

Question 11

Change the Cartesian integral to an equivalent polar integral, and then evaluate.
-$\int _ { - 3 } ^ { 3 } \int _ { - \sqrt { 9 - y ^ { 2 } } } ^ { 0 } \frac { \sqrt { x ^ { 2 } + y ^ { 2 } } } { 1 + \sqrt { x ^ { 2 } + y ^ { 2 } } } d x d y$

Accepted Answer

A)  $\frac { \pi ( 3 + 2 \ln 4 ) } { 4 }$ 
B)  $\frac { \pi ( 3 + 2 \ln 4 ) } { 2 }$ 
C)  $\frac { \pi ( 3 + \ln 4 ) } { 2 }$ 
D)  $\frac { \pi ( 3 + \ln 4 ) } { 4 }$

Question 12

Evaluate the integral.
-$$\int _ { 0 } ^ { 2 } \int _ { 0 } ^ { 8 } ( x + y ) d x d y$$

Accepted Answer

A)  5 
B)  480 
C)  80 
D)  30

Question 13

Solve the problem.
-Find the mass of a thin circular plate bounded by $x ^ { 2 } + y ^ { 2 } = 4$ if $\delta ( x , y ) = x ^ { 2 } + y ^ { 2 }$.

Accepted Answer

A)  $8 \pi$ 
B)  $2 \pi$ 
C)  $1 \pi$ 
D)  $4 \pi$

Question 14

Find the volume of the indicated region.
-the region under the surface $z = x ^ { 2 } + y ^ { 4 }$, and bounded by the planes $x = 0$ and $y = 16$ and the cylinder $y = x ^ { 2 }$

Accepted Answer

A)  $\frac { 7,886,848 } { 165 }$ 
B)  $\frac { 125,851,648 } { 165 }$ 
C)  $\frac { 1,988,608 } { 165 }$ 
D)  $\frac { 31,479,808 } { 165 }$

Question 15

Use a spherical coordinate integral to find the volume of the given solid.
-the solid between the spheres $\varrho = \cos \varphi$ and $\varrho = 6$

Accepted Answer

A)  $\frac { 863 } { 6 } \pi$ 
B)  $288 \pi$ 
C)  $\frac { 1727 } { 6 } \pi$ 
D)  $144 \pi$

Question 16

Solve the problem.
-Find the centroid of the region cut from the first quadrant by the line $x + y = 4$.

Accepted Answer

A)  $\bar { x } = \frac { 8 } { 3 } , \bar { y } = \frac { 8 } { 3 }$ 
B)  $\bar { x } = 1 , \bar { y } = 1$ 
C)  $\bar { x } = \frac { 4 } { 3 } , \bar { y } = \frac { 4 } { 3 }$ 
D)  $\bar { x } = 2 , \bar { y } = 2$

Question 17

Find the average value of the function f over the given region.
-$\mathrm { f } ( \mathrm { x } , \mathrm { y } ) = \mathrm { e } ^ { 10 \mathrm { x } }$ over the square $0 \leq \mathrm { x } \leq \frac { 1 } { 10 } , 0 \leq \mathrm { y } \leq \frac { 1 } { 10 }$.

Accepted Answer

A)  $2 \mathrm { e } - 1$ 
B)  e - 1 
C)  $\frac { 2 \mathrm { e } - 1 } { 100 }$ 
D)  $\frac { \mathrm { e } - 1 } { 10 }$

Question 18

Use a CAS integration utility to evaluate the triple integral of the given function over the specified solid region.
-F(x, y, z) = 9x + 8y + 10z over the tetrahedron bounded by the coordinate planes and the plane x + y + z = 1

Accepted Answer

A)  $\frac { 3 } { 2 }$ 
B)  $\frac { 3 } { 4 }$ 
C)  $\frac { 9 } { 8 }$ 
D)  1

Question 19

Solve the problem.
-Evaluate
$$\iiint _ { R } \int \sqrt { 1 - \left( \frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 16 } + \frac { z ^ { 2 } } { 9 } \right) } d x d y d z$$
where $R$ is the interior of the ellipsoid $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 16 } + \frac { z ^ { 2 } } { 9 } = 1$. Hint: Let $x = 5 u , y = 4 v , z = 3 w$, and then convert to spherical coordinates.

Accepted Answer

A)  $15 \pi$ 
B)  $20 \pi$ 
C)  $20 \pi ^ { 2 }$ 
D)  $15 \pi ^ { 2 }$

Question 20

Find the volume of the indicated region.
-the region enclosed by the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 100$ and the cylinder $( x - 5 ) ^ { 2 } + y ^ { 2 } = 25$

Accepted Answer

A)  $\frac { 2000 } { 9 } ( 3 \pi - 4 )$ 
B)  $\frac { 500 } { 3 } ( 3 \pi - 4 )$ 
C)  $\frac { 2000 } { 3 } ( 3 \pi - 4 )$ 
D)  $\frac { 625 } { 3 } ( 3 \pi - 4 )$

Exam 16: Multiple Integrals

Find the volume of the indicated region. -the region bounded below by the $x y$ -plane, laterally by the cylinder $r = 4 \cos \theta$ , and above by the plane $z = 3$

Find the volume of the indicated region. - $\text { the tetrahedron bounded by the coordinate planes and the plane } \frac { x } { 9 } + \frac { y } { 8 } + \frac { z } { 10 } = 1$

Evaluate the integral by changing the order of integration in an appropriate way. - $\int _ { 0 } ^ { 10 } \int _ { 1 } ^ { 2 } \int _ { \sqrt { x / 10 } } ^ { 1 } \frac { e ^ { y ^ { 3 } } } { z } d y d z d x$

Evaluate the integral. - $\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { 1 } ( 9 x - 8 y ) d y d x$

Evaluate the improper integral. - $\int _ { 0 } ^ { \infty } \int _ { 0 } ^ { \infty } \frac { d x d y } { ( x + 3 ) ^ { 2 } ( y + 10 ) ^ { 2 } }$

Change the Cartesian integral to an equivalent polar integral, and then evaluate. - $\int _ { - 3 } ^ { 3 } \int _ { - \sqrt { 9 - x ^ { 2 } } } ^ { \sqrt { 9 - x ^ { 2 } } } \frac { 1 } { \left( 1 + x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } } d y d x$

Evaluate the integral. - $\int _ { 0 } ^ { \ln 6 } \int _ { e ^ { y } } ^ { 6 } e ^ { y } d x d y$

Find the average value of F(x, y, z) over the given region. -F(x, y, z) = 9x + 10y + 3z over the rectangular solid in the first octant bounded by the coordinate planes and the planes x = 4, y = 7, z = 6

Change the Cartesian integral to an equivalent polar integral, and then evaluate. - $\int _ { - 3 } ^ { 3 } \int _ { - \sqrt { 9 - y ^ { 2 } } } ^ { 0 } \frac { \sqrt { x ^ { 2 } + y ^ { 2 } } } { 1 + \sqrt { x ^ { 2 } + y ^ { 2 } } } d x d y$

Evaluate the integral. - $\int _ { 0 } ^ { 2 } \int _ { 0 } ^ { 8 } ( x + y ) d x d y$

Solve the problem. -Find the mass of a thin circular plate bounded by $x ^ { 2 } + y ^ { 2 } = 4$ if $\delta ( x , y ) = x ^ { 2 } + y ^ { 2 }$ .

Find the volume of the indicated region. -the region under the surface $z = x ^ { 2 } + y ^ { 4 }$ , and bounded by the planes $x = 0$ and $y = 16$ and the cylinder $y = x ^ { 2 }$

Use a spherical coordinate integral to find the volume of the given solid. -the solid between the spheres $\varrho = \cos \varphi$ and $\varrho = 6$

Solve the problem. -Find the centroid of the region cut from the first quadrant by the line $x + y = 4$ .

Find the average value of the function f over the given region. - $\mathrm { f } ( \mathrm { x } , \mathrm { y } ) = \mathrm { e } ^ { 10 \mathrm { x } }$ over the square $0 \leq \mathrm { x } \leq \frac { 1 } { 10 } , 0 \leq \mathrm { y } \leq \frac { 1 } { 10 }$ .

Use a CAS integration utility to evaluate the triple integral of the given function over the specified solid region. -F(x, y, z) = 9x + 8y + 10z over the tetrahedron bounded by the coordinate planes and the plane x + y + z = 1

Find the volume of the indicated region. -the region enclosed by the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 100$ and the cylinder $( x - 5 ) ^ { 2 } + y ^ { 2 } = 25$

Exam 16: Multiple Integrals

Find the volume of the indicated region. -the region bounded below by the xyx yxy -plane, laterally by the cylinder r=4cos⁡θr = 4 \cos \thetar=4cosθ , and above by the plane z=3z = 3z=3

Evaluate the integral by changing the order of integration in an appropriate way. - ∫010∫12∫x/101ey3zdydzdx\int _ { 0 } ^ { 10 } \int _ { 1 } ^ { 2 } \int _ { \sqrt { x / 10 } } ^ { 1 } \frac { e ^ { y ^ { 3 } } } { z } d y d z d x∫010​∫12​∫x/10​1​zey3​dydzdx

Evaluate the integral. - ∫01∫01(9x−8y) dydx\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { 1 } ( 9 x - 8 y ) d y d x∫01​∫01​(9x−8y) dydx

Evaluate the improper integral. - ∫0∞∫0∞dxdy(x+3) 2(y+10) 2\int _ { 0 } ^ { \infty } \int _ { 0 } ^ { \infty } \frac { d x d y } { ( x + 3 ) ^ { 2 } ( y + 10 ) ^ { 2 } }∫0∞​∫0∞​(x+3) 2(y+10) 2dxdy​

Evaluate the integral. - ∫0ln⁡6∫ey6eydxdy\int _ { 0 } ^ { \ln 6 } \int _ { e ^ { y } } ^ { 6 } e ^ { y } d x d y∫0ln6​∫ey6​eydxdy

Find the average value of F(x, y, z) over the given region. -F(x, y, z) = 9x + 10y + 3z over the rectangular solid in the first octant bounded by the coordinate planes and the planes x = 4, y = 7, z = 6

Evaluate the integral. - ∫02∫08(x+y) dxdy\int _ { 0 } ^ { 2 } \int _ { 0 } ^ { 8 } ( x + y ) d x d y∫02​∫08​(x+y) dxdy

Solve the problem. -Find the mass of a thin circular plate bounded by x2+y2=4x ^ { 2 } + y ^ { 2 } = 4x2+y2=4 if δ(x,y) =x2+y2\delta ( x , y ) = x ^ { 2 } + y ^ { 2 }δ(x,y) =x2+y2 .

Find the volume of the indicated region. -the region under the surface z=x2+y4z = x ^ { 2 } + y ^ { 4 }z=x2+y4 , and bounded by the planes x=0x = 0x=0 and y=16y = 16y=16 and the cylinder y=x2y = x ^ { 2 }y=x2

Use a spherical coordinate integral to find the volume of the given solid. -the solid between the spheres ϱ=cos⁡φ\varrho = \cos \varphiϱ=cosφ and ϱ=6\varrho = 6ϱ=6

Solve the problem. -Find the centroid of the region cut from the first quadrant by the line x+y=4x + y = 4x+y=4 .