Question 1

Write an equivalent double integral with the order of integration reversed.
- ∫37∫10−x7dydx\int _ { 3 } ^ { 7 } \int _ { 10 - x } ^ { 7 } d y d x∫37​∫10−x7​dydx

Accepted Answer

A)    ∫34∫10−y7dxdy\int _ { 3 } ^ { 4 } \int _ { 10 - y } ^ { 7 } d x d y∫34​∫10−y7​dxdy 
B)    ∫37∫10−y7dxdy\int _ { 3 } ^ { 7 } \int _ { 10 - y } ^ { 7 } d x d y∫37​∫10−y7​dxdy 
C)    ∫37∫7−y4dxdy\int _ { 3 } ^ { 7 } \int _ { 7 - y } ^ { 4 } d x d y∫37​∫7−y4​dxdy 
D)    ∫34∫7−y4dxdy\int _ { 3 } ^ { 4 } \int _ { 7 - y } ^ { 4 } d x d y∫34​∫7−y4​dxdy

Question 2

Evaluate the integral by changing the order of integration in an appropriate way.
- ∫064∫07∫x34zy4+1dydzdx\int _ { 0 } ^ { 64 } \int _ { 0 } ^ { 7 } \int _ { \sqrt [ 3 ] { x } } ^ { 4 } \frac { z } { y ^ { 4 } + 1 } d y d z d x∫064​∫07​∫3x​4​y4+1z​dydzdx

Accepted Answer

A)    498ln⁡65\frac { 49 } { 8 } \ln 65849​ln65 
B)    498ln⁡257\frac { 49 } { 8 } \ln 257849​ln257 
C)    494ln⁡65\frac { 49 } { 4 } \ln 65449​ln65 
D)    494ln⁡257\frac { 49 } { 4 } \ln 257449​ln257

Question 3

Find the average value of F(x, y, z)  over the given region.
-F(x, y, z)  = 10x over the cube in the first octant bounded by the coordinate planes and the planes x = 5, y = 5, z = 5

Accepted Answer

A)   250
B)   1250
C)   25
D)   125

Question 4

Solve the problem.
-Find the moment of inertia about the z-axis of a thick-walled right circular cylinder bounded on the inside by the cylinder r = 1, on the outside by the cylinder r = 2, and on the top and bottom by the planes
Z = 7 and z = 11.

Accepted Answer

A)    30π30 \pi30π 
B)    15π15 \pi15π 
C)    32π32 \pi32π 
D)    60π60 \pi60π

Question 5

Evaluate the integral.
- ∫24∫4π10π∫0zrzdrdθdz\int _ { 2 } ^ { 4 } \int _ { 4 \pi } ^ { 10 \pi } \int _ { 0 } ^ { \mathrm { z } } \frac { \mathrm { r } } { \mathrm { z } } \mathrm { dr } \mathrm { d } 	heta \mathrm { dz }∫24​∫4π10π​∫0z​zr​drdθdz

Accepted Answer

A)    36π36 \pi36π 
B)    12π12 \pi12π 
C)    24π24 \pi24π 
D)    18π18 \pi18π

Question 6

Evaluate the integral.
- ∫0π/2∫0π/2∫02cos⁡φϱ2sin⁡φdφdθdφ\int _ { 0 } ^ { \pi / 2 } \int _ { 0 } ^ { \pi / 2 } \int _ { 0 } ^ { 2 \cos \varphi } \varrho ^ { 2 } \sin \varphi \mathrm { d } \varphi \mathrm { d } 	heta \mathrm { d } \varphi∫0π/2​∫0π/2​∫02cosφ​ϱ2sinφdφdθdφ

Accepted Answer

A)    49π2\frac { 4 } { 9 } \pi ^ { 2 }94​π2 
B)    13π\frac { 1 } { 3 } \pi31​π 
C)    49π\frac { 4 } { 9 } \pi94​π 
D)    13π2\frac { 1 } { 3 } \pi ^ { 2 }31​π2

Question 7

Find the volume of the indicated region.
-the region enclosed by the cone  z2=x2+y2z ^ { 2 } = x ^ { 2 } + y ^ { 2 }z2=x2+y2  between the planes  z=4z = 4z=4  and  z=10z = 10z=10

Accepted Answer

A)   234
B)    312π312 \pi312π 
C)    234π234 \pi234π 
D)   312

Question 8

Solve the problem.
-Find the centroid of the rectangular solid defined by  0≤x≤8,0≤y≤7,0≤z≤90 \leq x \leq 8,0 \leq y \leq 7,0 \leq z \leq 90≤x≤8,0≤y≤7,0≤z≤9 .

Accepted Answer

A)    xˉ=83,yˉ=73,zˉ=3\bar { x } = \frac { 8 } { 3 } , \bar { y } = \frac { 7 } { 3 } , \bar { z } = 3xˉ=38​,yˉ​=37​,zˉ=3 
B)    xˉ=8,yˉ=7,zˉ=9\bar { x } = 8 , \bar { y } = 7 , \bar { z } = 9xˉ=8,yˉ​=7,zˉ=9 
C)    xˉ=2,yˉ=74,zˉ=94\bar { x } = 2 , \bar { y } = \frac { 7 } { 4 } , \bar { z } = \frac { 9 } { 4 }xˉ=2,yˉ​=47​,zˉ=49​ 
D)    xˉ=4,yˉ=72,zˉ=92\bar { x } = 4 , \bar { y } = \frac { 7 } { 2 } , \bar { z } = \frac { 9 } { 2 }xˉ=4,yˉ​=27​,zˉ=29​

Question 9

Express the area of the region bounded by the given line(s)  and/or curve(s)  as an iterated double integral.
-The parabola  y=x2y = x ^ { 2 }y=x2  and the line  y=−8x+20y = - 8 x + 20y=−8x+20

Accepted Answer

A)    ∫02∫20−8x+x2dydx\int _ { 0 } ^ { 2 } \int _ { 20 } ^ { - 8 x + x ^ { 2 } } d y d x∫02​∫20−8x+x2​dydx 
B)    ∫−102∫x2−8x+20dydx\int _ { - 10 } ^ { 2 } \int _ { x ^ { 2 } } ^ { - 8 x + 20 } d y d x∫−102​∫x2−8x+20​dydx 
C)    ∫−102∫0−8x+20−x2dxdy\int _ { - 10 } ^ { 2 } \int _ { 0 } ^ { - 8 x + 20 - x ^ { 2 } } d x d y∫−102​∫0−8x+20−x2​dxdy 
D)    ∫02∫x2−8x+20dydx\int _ { 0 } ^ { 2 } \int _ { x ^ { 2 } } ^ { - 8 x + 20 } d y d x∫02​∫x2−8x+20​dydx

Question 10

Solve the problem.
-Find the moment of inertia about the  yyy -axis of the thin infinite region of constant density  δ=3\delta = 3δ=3  in the first quadrant bounded by the coordinate axes and the curve  y=e−7xy = e ^ { - 7 x }y=e−7x .

Accepted Answer

A)    9343\frac { 9 } { 343 }3439​ 
B)    1343\frac { 1 } { 343 }3431​ 
C)    6343\frac { 6 } { 343 }3436​ 
D)    3686\frac { 3 } { 686 }6863​

Question 11

Use a spherical coordinate integral to find the volume of the given solid.
-the solid between the spheres  ϱ=3cos⁡φ\varrho = 3 \cos \varphiϱ=3cosφ  and  ϱ=7cos⁡φ\varrho = 7 \cos \varphiϱ=7cosφ

Accepted Answer

A)    3163π\frac { 316 } { 3 } \pi3316​π 
B)    158π158 \pi158π 
C)    79π79 \pi79π 
D)    1583π\frac { 158 } { 3 } \pi3158​π

Question 12

Provide an appropriate response.
-x = 7u cos 10v, y = 7u sin 10v, z = 3w

Accepted Answer

A)   2100u
B)   1470u
C)   2100v
D)   1470v

Question 13

Use the given transformation to evaluate the integral.
- u=2x+y−z,v=−x+y+z,w=−x+y+2z∭(2x+y−z) (z−x+y) dxdydz\begin{aligned}u = & 2 x + y - z , v = - x + y + z , w = - x + y + 2 z \& \iiint ( 2 x + y - z )  ( z - x + y )  d x d y d z\end{aligned}u=​2x+y−z,v=−x+y+z,w=−x+y+2z∭(2x+y−z) (z−x+y) dxdydz​ 
where  RRR  is the parallelepiped bounded by the planes  2x+y−z=2,2x+y−z=6,−x+y+z=3,−x+y+z=42 x + y - z = 2,2 x + y - z = 6 , - x + y + z = 3 , - x + y + z = 42x+y−z=2,2x+y−z=6,−x+y+z=3,−x+y+z=4 ,  −x+y+2z=6,−x+y+2z=8- x + y + 2 z = 6 , - x + y + 2 z = 8−x+y+2z=6,−x+y+2z=8

Accepted Answer

A)    2243\frac { 224 } { 3 }3224​ 
B)    1123\frac { 112 } { 3 }3112​ 
C)   336
D)   672

Question 14

Change the Cartesian integral to an equivalent polar integral, and then evaluate.
- ∫−90∫−81−x2011+x2+y2dydx\int _ { - 9 } ^ { 0 } \int _ { - \sqrt { 81 - x ^ { 2 } } } ^ { 0 } \frac { 1 } { 1 + \sqrt { x ^ { 2 } + y ^ { 2 } } } d y d x∫−90​∫−81−x2​0​1+x2+y2​1​dydx

Accepted Answer

A)    π(9+ln⁡10) 2\frac { \pi ( 9 + \ln 10 )  } { 2 }2π(9+ln10) ​ 
B)    π(9+ln⁡10) 4\frac { \pi ( 9 + \ln 10 )  } { 4 }4π(9+ln10) ​ 
C)    π(9−ln⁡10) 4\frac { \pi ( 9 - \ln 10 )  } { 4 }4π(9−ln10) ​ 
D)    π(9−ln⁡10) 2\frac { \pi ( 9 - \ln 10 )  } { 2 }2π(9−ln10) ​

Question 15

Evaluate the improper integral.
- ∫036∫0100dxdyxy\int _ { 0 } ^ { 36 } \int _ { 0 } ^ { 100 } \frac { d x d y } { \sqrt { x y } }∫036​∫0100​xy​dxdy​

Accepted Answer

A)   40
B)   240
C)   24
D)   360

Question 16

Solve the problem.
-Find the center of mass of a thin infinite region in the first quadrant bounded by the coordinate axes and the curve  y=e−4xy = e ^ { - 4 x }y=e−4x  if  δ(x,y) =xy\delta ( x , y )  = x yδ(x,y) =xy .

Accepted Answer

A)    xˉ=16,yˉ=29\bar { x } = \frac { 1 } { 6 } , \bar { y } = \frac { 2 } { 9 }xˉ=61​,yˉ​=92​ 
B)    xˉ=14,yˉ=827\bar { x } = \frac { 1 } { 4 } , \bar { y } = \frac { 8 } { 27 }xˉ=41​,yˉ​=278​ 
C)    xˉ=14,yˉ=29\bar { x } = \frac { 1 } { 4 } , \bar { y } = \frac { 2 } { 9 }xˉ=41​,yˉ​=92​ 
D)    xˉ=16,yˉ=827\bar { x } = \frac { 1 } { 6 } , \bar { y } = \frac { 8 } { 27 }xˉ=61​,yˉ​=278​

Question 17

Solve the problem.
-Evaluate
 ∫−22∫−4−y24−y2∫09−x2−y2dzdxdy\int _ { - 2 } ^ { 2 } \int _ { - \sqrt { 4 - y ^ { 2 } } } ^ { \sqrt { 4 - y ^ { 2 } } } \int _ { 0 } ^ { \sqrt { 9 - x ^ { 2 } - y ^ { 2 } } } d z d x d y∫−22​∫−4−y2​4−y2​​∫09−x2−y2​​dzdxdy 
by transforming to cylindrical or spherical coordinates.

Accepted Answer

A)    2π3(9−(5) 3/2) \frac { 2 \pi } { 3 } \left( 9 - ( 5 )  ^ { 3 / 2 } \right) 32π​(9−(5) 3/2)  
B)    2π3(27−(5) 3/2) \frac { 2 \pi } { 3 } \left( 27 - ( 5 )  ^ { 3 / 2 } \right) 32π​(27−(5) 3/2)  
C)    4π3(9−(5) 3/2) \frac { 4 \pi } { 3 } \left( 9 - ( 5 )  ^ { 3 / 2 } \right) 34π​(9−(5) 3/2)  
D)    4π3(27−(5) 3/2) \frac { 4 \pi } { 3 } \left( 27 - ( 5 )  ^ { 3 / 2 } \right) 34π​(27−(5) 3/2)

Question 18

Evaluate the integral by changing the order of integration in an appropriate way.
- ∫050∫x/105∫0∞ze−(y2+z2) dzdydx\int _ { 0 } ^ { 50 } \int _ { x / 10 } ^ { 5 } \int _ { 0 } ^ { \infty } z e ^ { - \left( y ^ { 2 } + z ^ { 2 } \right)  } d z d y d x∫050​∫x/105​∫0∞​ze−(y2+z2) dzdydx

Accepted Answer

A)    5(1−e−50) 5 \left( 1 - e ^ { - 50 } \right) 5(1−e−50)  
B)    5(1−e−25) 5 \left( 1 - \mathrm { e } ^ { - 25 } \right) 5(1−e−25)  
C)    52(1−e−50) \frac { 5 } { 2 } \left( 1 - \mathrm { e } ^ { - 50 } \right) 25​(1−e−50)  
D)    52(1−e−25) \frac { 5 } { 2 } \left( 1 - \mathrm { e } ^ { - 25 } \right) 25​(1−e−25)

Question 19

Find the average value of the function f over the given region.
-f(x, y)  = 8x + 5y over the triangle with vertices (0, 0) , (2, 0) , and (0, 7) .

Accepted Answer

A)    233\frac { 23 } { 3 }323​ 
B)   7
C)    163\frac { 16 } { 3 }316​ 
D)   17

Question 20

Find the volume of the indicated region.
-the region bounded by the cylinders r = 5, r = 8 and the planes z = 7, z = 10

Accepted Answer

A)    234π234 \pi234π 
B)    468π468 \pi468π 
C)    351π351 \pi351π 
D)    117π117 \pi117π

Exam 16: Multiple Integrals

Find the average value of the function f over the given region. -f(x, y) = 8x + 5y over the triangle with vertices (0, 0) , (2, 0) , and (0, 7) .

Correct Answer
verified

Solve the problem. -Evaluate $\int _ { - 2 } ^ { 2 } \int _ { - \sqrt { 4 - y ^ { 2 } } } ^ { \sqrt { 4 - y ^ { 2 } } } \int _ { 0 } ^ { \sqrt { 9 - x ^ { 2 } - y ^ { 2 } } } d z d x d y$ by transforming to cylindrical or spherical coordinates.

Correct Answer
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Solve the problem. -Find the centroid of the rectangular solid defined by $0 \leq x \leq 8,0 \leq y \leq 7,0 \leq z \leq 9$ .

Correct Answer
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Provide an appropriate response. -x = 7u cos 10v, y = 7u sin 10v, z = 3w

Correct Answer
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Evaluate the integral. - $\int _ { 0 } ^ { \pi / 2 } \int _ { 0 } ^ { \pi / 2 } \int _ { 0 } ^ { 2 \cos \varphi } \varrho ^ { 2 } \sin \varphi \mathrm { d } \varphi \mathrm { d } \theta \mathrm { d } \varphi$

Correct Answer
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Solve the problem. -Find the center of mass of a thin infinite region in the first quadrant bounded by the coordinate axes and the curve $y = e ^ { - 4 x }$ if $\delta ( x , y ) = x y$ .

Correct Answer
verified

Evaluate the integral by changing the order of integration in an appropriate way. - $\int _ { 0 } ^ { 50 } \int _ { x / 10 } ^ { 5 } \int _ { 0 } ^ { \infty } z e ^ { - \left( y ^ { 2 } + z ^ { 2 } \right) } d z d y d x$

Correct Answer
verified

Change the Cartesian integral to an equivalent polar integral, and then evaluate. - $\int _ { - 9 } ^ { 0 } \int _ { - \sqrt { 81 - x ^ { 2 } } } ^ { 0 } \frac { 1 } { 1 + \sqrt { x ^ { 2 } + y ^ { 2 } } } d y d x$

Correct Answer
verified

Evaluate the integral by changing the order of integration in an appropriate way. - $\int _ { 0 } ^ { 64 } \int _ { 0 } ^ { 7 } \int _ { \sqrt [ 3 ] { x } } ^ { 4 } \frac { z } { y ^ { 4 } + 1 } d y d z d x$

Correct Answer
verified

Find the volume of the indicated region. -the region bounded by the cylinders r = 5, r = 8 and the planes z = 7, z = 10

Correct Answer
verified

Use a spherical coordinate integral to find the volume of the given solid. -the solid between the spheres $\varrho = 3 \cos \varphi$ and $\varrho = 7 \cos \varphi$

Correct Answer
verified

Write an equivalent double integral with the order of integration reversed. - $\int _ { 3 } ^ { 7 } \int _ { 10 - x } ^ { 7 } d y d x$

Correct Answer
verified

Find the average value of F(x, y, z) over the given region. -F(x, y, z) = 10x over the cube in the first octant bounded by the coordinate planes and the planes x = 5, y = 5, z = 5

Correct Answer
verified

Evaluate the improper integral. - $\int _ { 0 } ^ { 36 } \int _ { 0 } ^ { 100 } \frac { d x d y } { \sqrt { x y } }$

Correct Answer
verified

Correct Answer
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Solve the problem. -Find the moment of inertia about the z-axis of a thick-walled right circular cylinder bounded on the inside by the cylinder r = 1, on the outside by the cylinder r = 2, and on the top and bottom by the planes Z = 7 and z = 11.

Correct Answer
verified

Evaluate the integral. - $\int _ { 2 } ^ { 4 } \int _ { 4 \pi } ^ { 10 \pi } \int _ { 0 } ^ { \mathrm { z } } \frac { \mathrm { r } } { \mathrm { z } } \mathrm { dr } \mathrm { d } \theta \mathrm { dz }$

Correct Answer
verified

Solve the problem. -Find the moment of inertia about the $y$ -axis of the thin infinite region of constant density $\delta = 3$ in the first quadrant bounded by the coordinate axes and the curve $y = e ^ { - 7 x }$ .

Correct Answer
verified

Express the area of the region bounded by the given line(s) and/or curve(s) as an iterated double integral. -The parabola $y = x ^ { 2 }$ and the line $y = - 8 x + 20$

Correct Answer
verified

Find the volume of the indicated region. -the region enclosed by the cone $z ^ { 2 } = x ^ { 2 } + y ^ { 2 }$ between the planes $z = 4$ and $z = 10$

Correct Answer
verified

Exam 16: Multiple Integrals

Find the average value of the function f over the given region. -f(x, y) = 8x + 5y over the triangle with vertices (0, 0) , (2, 0) , and (0, 7) .

Correct AnswerverifiedShow Answer

Correct AnswerverifiedShow Answer

Solve the problem. -Find the centroid of the rectangular solid defined by 0≤x≤8,0≤y≤7,0≤z≤90 \leq x \leq 8,0 \leq y \leq 7,0 \leq z \leq 90≤x≤8,0≤y≤7,0≤z≤9 .

Correct AnswerverifiedShow Answer

Provide an appropriate response. -x = 7u cos 10v, y = 7u sin 10v, z = 3w

Correct AnswerverifiedShow Answer

Correct AnswerverifiedShow Answer

Solve the problem. -Find the center of mass of a thin infinite region in the first quadrant bounded by the coordinate axes and the curve y=e−4xy = e ^ { - 4 x }y=e−4x if δ(x,y) =xy\delta ( x , y ) = x yδ(x,y) =xy .

Correct AnswerverifiedShow Answer

Correct AnswerverifiedShow Answer

Change the Cartesian integral to an equivalent polar integral, and then evaluate. - ∫−90∫−81−x2011+x2+y2dydx\int _ { - 9 } ^ { 0 } \int _ { - \sqrt { 81 - x ^ { 2 } } } ^ { 0 } \frac { 1 } { 1 + \sqrt { x ^ { 2 } + y ^ { 2 } } } d y d x∫−90​∫−81−x2​0​1+x2+y2​1​dydx

Correct AnswerverifiedShow Answer

Evaluate the integral by changing the order of integration in an appropriate way. - ∫064∫07∫x34zy4+1dydzdx\int _ { 0 } ^ { 64 } \int _ { 0 } ^ { 7 } \int _ { \sqrt [ 3 ] { x } } ^ { 4 } \frac { z } { y ^ { 4 } + 1 } d y d z d x∫064​∫07​∫3x​4​y4+1z​dydzdx

Correct AnswerverifiedShow Answer

Find the volume of the indicated region. -the region bounded by the cylinders r = 5, r = 8 and the planes z = 7, z = 10

Correct AnswerverifiedShow Answer

Use a spherical coordinate integral to find the volume of the given solid. -the solid between the spheres ϱ=3cos⁡φ\varrho = 3 \cos \varphiϱ=3cosφ and ϱ=7cos⁡φ\varrho = 7 \cos \varphiϱ=7cosφ

Correct AnswerverifiedShow Answer

Write an equivalent double integral with the order of integration reversed. - ∫37∫10−x7dydx\int _ { 3 } ^ { 7 } \int _ { 10 - x } ^ { 7 } d y d x∫37​∫10−x7​dydx

Correct AnswerverifiedShow Answer

Find the average value of F(x, y, z) over the given region. -F(x, y, z) = 10x over the cube in the first octant bounded by the coordinate planes and the planes x = 5, y = 5, z = 5

Correct AnswerverifiedShow Answer

Evaluate the improper integral. - ∫036∫0100dxdyxy\int _ { 0 } ^ { 36 } \int _ { 0 } ^ { 100 } \frac { d x d y } { \sqrt { x y } }∫036​∫0100​xy​dxdy​

Correct AnswerverifiedShow Answer

Correct AnswerverifiedShow Answer

Solve the problem. -Find the moment of inertia about the z-axis of a thick-walled right circular cylinder bounded on the inside by the cylinder r = 1, on the outside by the cylinder r = 2, and on the top and bottom by the planes Z = 7 and z = 11.

Correct AnswerverifiedShow Answer

Evaluate the integral. - ∫24∫4π10π∫0zrzdrdθdz\int _ { 2 } ^ { 4 } \int _ { 4 \pi } ^ { 10 \pi } \int _ { 0 } ^ { \mathrm { z } } \frac { \mathrm { r } } { \mathrm { z } } \mathrm { dr } \mathrm { d } \theta \mathrm { dz }∫24​∫4π10π​∫0z​zr​drdθdz

Correct AnswerverifiedShow Answer

Solve the problem. -Find the moment of inertia about the yyy -axis of the thin infinite region of constant density δ=3\delta = 3δ=3 in the first quadrant bounded by the coordinate axes and the curve y=e−7xy = e ^ { - 7 x }y=e−7x .

Correct AnswerverifiedShow Answer

Express the area of the region bounded by the given line(s) and/or curve(s) as an iterated double integral. -The parabola y=x2y = x ^ { 2 }y=x2 and the line y=−8x+20y = - 8 x + 20y=−8x+20

Correct AnswerverifiedShow Answer

Find the volume of the indicated region. -the region enclosed by the cone z2=x2+y2z ^ { 2 } = x ^ { 2 } + y ^ { 2 }z2=x2+y2 between the planes z=4z = 4z=4 and z=10z = 10z=10

Correct AnswerverifiedShow Answer

Correct Answer
verified

Correct Answer
verified

Solve the problem. -Find the centroid of the rectangular solid defined by $0 \leq x \leq 8,0 \leq y \leq 7,0 \leq z \leq 9$ .

Correct Answer
verified

Correct Answer
verified

Correct Answer
verified

Solve the problem. -Find the center of mass of a thin infinite region in the first quadrant bounded by the coordinate axes and the curve $y = e ^ { - 4 x }$ if $\delta ( x , y ) = x y$ .

Correct Answer
verified

Correct Answer
verified

Change the Cartesian integral to an equivalent polar integral, and then evaluate. - $\int _ { - 9 } ^ { 0 } \int _ { - \sqrt { 81 - x ^ { 2 } } } ^ { 0 } \frac { 1 } { 1 + \sqrt { x ^ { 2 } + y ^ { 2 } } } d y d x$

Correct Answer
verified

Evaluate the integral by changing the order of integration in an appropriate way. - $\int _ { 0 } ^ { 64 } \int _ { 0 } ^ { 7 } \int _ { \sqrt [ 3 ] { x } } ^ { 4 } \frac { z } { y ^ { 4 } + 1 } d y d z d x$

Correct Answer
verified

Correct Answer
verified

Use a spherical coordinate integral to find the volume of the given solid. -the solid between the spheres $\varrho = 3 \cos \varphi$ and $\varrho = 7 \cos \varphi$

Correct Answer
verified

Write an equivalent double integral with the order of integration reversed. - $\int _ { 3 } ^ { 7 } \int _ { 10 - x } ^ { 7 } d y d x$

Correct Answer
verified

Correct Answer
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Evaluate the improper integral. - $\int _ { 0 } ^ { 36 } \int _ { 0 } ^ { 100 } \frac { d x d y } { \sqrt { x y } }$

Correct Answer
verified

Correct Answer
verified

Correct Answer
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Evaluate the integral. - $\int _ { 2 } ^ { 4 } \int _ { 4 \pi } ^ { 10 \pi } \int _ { 0 } ^ { \mathrm { z } } \frac { \mathrm { r } } { \mathrm { z } } \mathrm { dr } \mathrm { d } \theta \mathrm { dz }$

Correct Answer
verified

Solve the problem. -Find the moment of inertia about the $y$ -axis of the thin infinite region of constant density $\delta = 3$ in the first quadrant bounded by the coordinate axes and the curve $y = e ^ { - 7 x }$ .

Correct Answer
verified

Express the area of the region bounded by the given line(s) and/or curve(s) as an iterated double integral. -The parabola $y = x ^ { 2 }$ and the line $y = - 8 x + 20$

Correct Answer
verified

Find the volume of the indicated region. -the region enclosed by the cone $z ^ { 2 } = x ^ { 2 } + y ^ { 2 }$ between the planes $z = 4$ and $z = 10$

Correct Answer
verified