Question 1

Find the volume of the solid bounded in the first octanat bounded by the cylinder $z = 9 - y ^ { 2 }$ and the planes $x = 1$.

Accepted Answer

The answer of Find the volume of the solid bounded...

Question 2

Evaluate $\iiint _ { T } f ( x , y , z ) d V$ where $f ( x , y , z ) = 7 y$ and $T$ is the region bounded by the paraboloid $y = x ^ { 2 } + z ^ { 2 }$ and the plane $y = 1$.

Accepted Answer

The answer of Evaluate \(\iiint _ { T } f...

Question 3

Evaluate the iterated integral.
$$\int _ { 1 } ^ { 3 } \int _ { y } ^ { 3 } 10 x y d x d y$$

Accepted Answer

A)  85 
B)  80 
C)  100 
D)  110 
E)  90 
A)  85 
B)  80 
C)  100 
D)  110 
E)  90

Question 4

Find the area of the part of hyperbolic paraboloid $z = y ^ { 2 } - x ^ { 2 }$ that lies between the cylinders $x ^ { 2 } + y ^ { 2 } = 1$ and $x ^ { 2 } + y ^ { 2 } = 9$.

Accepted Answer

The answer of Find the area of the part of...

Question 5

Find the mass of the lamina that occupies the region $D$ and has the given density function, if $D$ is bounded by the parabola $x = y ^ { 2 }$ and the line $y = x - 2$.
$$\rho ( x , y ) = 3$$

Accepted Answer

The answer of Find the mass of the lamina that...

Question 6

Calculate the iterated integral.
$$\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { y } 5 \cos \left( y ^ { 2 } \right) d x d y$$

Accepted Answer

The answer of Calculate the iterated integral.
\[\int _ { 0...

Question 7

For which of the following regions would you use rectangular coordinates?

Accepted Answer

A) 
  
B) 
  
C) 
  
A) 
  
B) 
  
C)

Question 8

$$\text { Find the area of the surface. The part of the sphere } x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 64 \text { that lies above the plane } z = 1 \text {. }$$

Accepted Answer

The answer of \[\text { Find the area of the...

Question 9

Find the area of the part of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 36 z$ that lies inside the paraboloid $z = x ^ { 2 } + y ^ { 2 }$.
Select the correct answer.

Accepted Answer

A)  $6 \pi$ 
B)  $36 \pi$ 
C)  $12.5 \pi$ 
D)  $16 \pi$ 
E)  $10.5 \pi$ 
A)  $6 \pi$ 
B)  $36 \pi$ 
C)  $12.5 \pi$ 
D)  $16 \pi$ 
E)  $10.5 \pi$

Question 10

Calculate the double integral.
$$\iint _ { R } \left( 9 x ^ { 2 } y ^ { 3 } - 15 x ^ { 4 } \right) d A , R = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq 4 \}$$

Accepted Answer

The answer of Calculate the double integral.
\[\iint _ { R...

Question 11

$$\text { Find the area of the surface. The part of the sphere } x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 64 \text { that lies above the plane } z = 1 \text {. }$$

Accepted Answer

The answer of \[\text { Find the area of the...

Question 12

An electric charge is spread over a rectangular region $R = \{ ( x , y ) \mid 0 \leq x \leq 3,0 \leq y \leq 4 \}$. Find the total charge on $R$ if the charge density at a point $( x , y )$ in $R$ (measured in coulombs per square meter $)$ is $\sigma ( x , y ) = x ^ { 2 } + 4 y ^ { 3 }$.
Select the correct answer.

Accepted Answer

A)  804 coulombs 
B)  91 coulombs 
C)  300 coulombs 
D)  265 coulombs 
A)  804 coulombs 
B)  91 coulombs 
C)  300 coulombs 
D)  265 coulombs

Question 13

Calculate the iterated integral.
$$\int _ { - 1 } ^ { 1 } \int _ { 0 } ^ { 1 } 5 y e ^ { x y } d x d y$$

Accepted Answer

The answer of Calculate the iterated integral.
\[\int _ { -...

Question 14

Use cylindrical or spherical coordinates, whichever seems more appropriate, to evaluate $\iiint _ { E } z d V$ where $E$ lies above the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ and below the plane $z = 25$.

Accepted Answer

The answer of Use cylindrical or spherical coordinates, whichever seems...

Question 15

An electric charge is spread over a rectangular region $R = \{ ( x , y ) \mid 0 \leq x \leq 3,0 \leq y \leq 2 \}$. Find the total charge on $R$ if the charge density at a point $( x , y )$ in $R$ (measured in coulombs per square meter) is $\sigma ( x , y ) = 4 x ^ { 2 } + y ^ { 3 }$.

Accepted Answer

A)  84 coulombs 
B)  222 coulombs 
C)  22 coulombs 
D)  56 coulombs 
A)  84 coulombs 
B)  222 coulombs 
C)  22 coulombs 
D)  56 coulombs

Question 16

Sketch the solid bounded by the graphs of the equations $z = x ^ { 2 } + y ^ { 2 }$ and $z = 8 - x ^ { 2 } - y ^ { 2 }$, and then use a triple integral to find the volume of the solid.

Accepted Answer

The answer of Sketch the solid bounded by the graphs...

Question 17

Use a triple integral to find the volume of the solid bounded by $x = y ^ { 2 }$ and the planes $z = 0$ and $x + z = 3$.
Select the correct answer.

Accepted Answer

A)  $8.3$ 
B)  $18.3$ 
C)  $2.5$ 
D)  $15.3$ 
E)  $11.3$ 
A)  $8.3$ 
B)  $18.3$ 
C)  $2.5$ 
D)  $15.3$ 
E)  $11.3$

Question 18

Find the center of mass of the lamina that occupies the region $D$ and has the given density function, if $D$ is bounded by the parabola $y = 1 - x ^ { 2 }$ and the $x$-axis.
$$\rho ( x , y ) = 4 y$$

Accepted Answer

A)  $( 8,4 )$ 
B)  $( 4,46.44 )$ 
C)  $( 0,0.57 )$ 
D)  $( 12,0.57 )$ 
E)  None of these 
A)  $( 8,4 )$ 
B)  $( 4,46.44 )$ 
C)  $( 0,0.57 )$ 
D)  $( 12,0.57 )$ 
E)  None of these

Question 19

Evaluate the iterated integral.
Select the correct answer.
$\int _ { 1 } ^ { 3 } \int _ { y } ^ { 3 } 12 x y d x d y$

Accepted Answer

A)  96 
B)  116 
C)  106 
D)  101 
E)  126 
A)  96 
B)  116 
C)  106 
D)  101 
E)  126

Question 20

$$\text { Evaluate } \iint _ { D } 4 x ^ { 2 } y ^ { 2 } d A \text { where } D \text { is the figure bounded by } y = 1 , y = 2 , x = 0 \text { and } x = y \text {. }$$

Accepted Answer

The answer of \[\text { Evaluate } \iint _ {...

Find the volume of the solid bounded in the first octanat bounded by the cylinder $z = 9 - y ^ { 2 }$ and the planes $x = 1$ .

Evaluate $\iiint _ { T } f ( x , y , z ) d V$ where $f ( x , y , z ) = 7 y$ and $T$ is the region bounded by the paraboloid $y = x ^ { 2 } + z ^ { 2 }$ and the plane $y = 1$ .

Evaluate the iterated integral. $\int _ { 1 } ^ { 3 } \int _ { y } ^ { 3 } 10 x y d x d y$

Find the area of the part of hyperbolic paraboloid $z = y ^ { 2 } - x ^ { 2 }$ that lies between the cylinders $x ^ { 2 } + y ^ { 2 } = 1$ and $x ^ { 2 } + y ^ { 2 } = 9$ .

Find the mass of the lamina that occupies the region $D$ and has the given density function, if $D$ is bounded by the parabola $x = y ^ { 2 }$ and the line $y = x - 2$ . $\rho ( x , y ) = 3$

Calculate the iterated integral. $\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { y } 5 \cos \left( y ^ { 2 } \right) d x d y$

For which of the following regions would you use rectangular coordinates?

$\text { Find the area of the surface. The part of the sphere } x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 64 \text { that lies above the plane } z = 1 \text {. }$

Find the area of the part of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 36 z$ that lies inside the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ . Select the correct answer.

Calculate the double integral. $\iint _ { R } \left( 9 x ^ { 2 } y ^ { 3 } - 15 x ^ { 4 } \right) d A , R = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq 4 \}$

$\text { Find the area of the surface. The part of the sphere } x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 64 \text { that lies above the plane } z = 1 \text {. }$

Calculate the iterated integral. $\int _ { - 1 } ^ { 1 } \int _ { 0 } ^ { 1 } 5 y e ^ { x y } d x d y$

Use cylindrical or spherical coordinates, whichever seems more appropriate, to evaluate $\iiint _ { E } z d V$ where $E$ lies above the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ and below the plane $z = 25$ .

An electric charge is spread over a rectangular region $R = \{ ( x , y ) \mid 0 \leq x \leq 3,0 \leq y \leq 2 \}$ . Find the total charge on $R$ if the charge density at a point $( x , y )$ in $R$ (measured in coulombs per square meter) is $\sigma ( x , y ) = 4 x ^ { 2 } + y ^ { 3 }$ .

Sketch the solid bounded by the graphs of the equations $z = x ^ { 2 } + y ^ { 2 }$ and $z = 8 - x ^ { 2 } - y ^ { 2 }$ , and then use a triple integral to find the volume of the solid.

Use a triple integral to find the volume of the solid bounded by $x = y ^ { 2 }$ and the planes $z = 0$ and $x + z = 3$ . Select the correct answer.

Find the center of mass of the lamina that occupies the region $D$ and has the given density function, if $D$ is bounded by the parabola $y = 1 - x ^ { 2 }$ and the $x$ -axis. $\rho ( x , y ) = 4 y$

Evaluate the iterated integral. Select the correct answer. $\int _ { 1 } ^ { 3 } \int _ { y } ^ { 3 } 12 x y d x d y$

$\text { Evaluate } \iint _ { D } 4 x ^ { 2 } y ^ { 2 } d A \text { where } D \text { is the figure bounded by } y = 1 , y = 2 , x = 0 \text { and } x = y \text {. }$

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Exam 15: Multiple Integrals

Find the volume of the solid bounded in the first octanat bounded by the cylinder z=9−y2z = 9 - y ^ { 2 }z=9−y2 and the planes x=1x = 1x=1 .

Evaluate ∭Tf(x,y,z)dV\iiint _ { T } f ( x , y , z ) d V∭T​f(x,y,z)dV where f(x,y,z)=7yf ( x , y , z ) = 7 yf(x,y,z)=7y and TTT is the region bounded by the paraboloid y=x2+z2y = x ^ { 2 } + z ^ { 2 }y=x2+z2 and the plane y=1y = 1y=1 .

Evaluate the iterated integral. ∫13∫y310xydxdy\int _ { 1 } ^ { 3 } \int _ { y } ^ { 3 } 10 x y d x d y∫13​∫y3​10xydxdy

Find the area of the part of hyperbolic paraboloid z=y2−x2z = y ^ { 2 } - x ^ { 2 }z=y2−x2 that lies between the cylinders x2+y2=1x ^ { 2 } + y ^ { 2 } = 1x2+y2=1 and x2+y2=9x ^ { 2 } + y ^ { 2 } = 9x2+y2=9 .

Find the mass of the lamina that occupies the region DDD and has the given density function, if DDD is bounded by the parabola x=y2x = y ^ { 2 }x=y2 and the line y=x−2y = x - 2y=x−2 . ρ(x,y)=3\rho ( x , y ) = 3ρ(x,y)=3

Calculate the iterated integral. ∫01∫0y5cos⁡(y2)dxdy\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { y } 5 \cos \left( y ^ { 2 } \right) d x d y∫01​∫0y​5cos(y2)dxdy

For which of the following regions would you use rectangular coordinates?

Find the area of the part of the sphere x2+y2+z2=36zx ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 36 zx2+y2+z2=36z that lies inside the paraboloid z=x2+y2z = x ^ { 2 } + y ^ { 2 }z=x2+y2 . Select the correct answer.

Calculate the double integral. ∬R(9x2y3−15x4)dA,R={(x,y)∣0≤x≤1,0≤y≤4}\iint _ { R } \left( 9 x ^ { 2 } y ^ { 3 } - 15 x ^ { 4 } \right) d A , R = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq 4 \}∬R​(9x2y3−15x4)dA,R={(x,y)∣0≤x≤1,0≤y≤4}

Calculate the iterated integral. ∫−11∫015yexydxdy\int _ { - 1 } ^ { 1 } \int _ { 0 } ^ { 1 } 5 y e ^ { x y } d x d y∫−11​∫01​5yexydxdy

Use cylindrical or spherical coordinates, whichever seems more appropriate, to evaluate ∭EzdV\iiint _ { E } z d V∭E​zdV where EEE lies above the paraboloid z=x2+y2z = x ^ { 2 } + y ^ { 2 }z=x2+y2 and below the plane z=25z = 25z=25 .

Sketch the solid bounded by the graphs of the equations z=x2+y2z = x ^ { 2 } + y ^ { 2 }z=x2+y2 and z=8−x2−y2z = 8 - x ^ { 2 } - y ^ { 2 }z=8−x2−y2 , and then use a triple integral to find the volume of the solid.

Use a triple integral to find the volume of the solid bounded by x=y2x = y ^ { 2 }x=y2 and the planes z=0z = 0z=0 and x+z=3x + z = 3x+z=3 . Select the correct answer.

Find the center of mass of the lamina that occupies the region DDD and has the given density function, if DDD is bounded by the parabola y=1−x2y = 1 - x ^ { 2 }y=1−x2 and the xxx -axis. ρ(x,y)=4y\rho ( x , y ) = 4 yρ(x,y)=4y

Evaluate the iterated integral. Select the correct answer. ∫13∫y312xydxdy\int _ { 1 } ^ { 3 } \int _ { y } ^ { 3 } 12 x y d x d y∫13​∫y3​12xydxdy

Functions and Models

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Techniques of Integration

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Find the volume of the solid bounded in the first octanat bounded by the cylinder $z = 9 - y ^ { 2 }$ and the planes $x = 1$ .

Evaluate $\iiint _ { T } f ( x , y , z ) d V$ where $f ( x , y , z ) = 7 y$ and $T$ is the region bounded by the paraboloid $y = x ^ { 2 } + z ^ { 2 }$ and the plane $y = 1$ .

Evaluate the iterated integral. $\int _ { 1 } ^ { 3 } \int _ { y } ^ { 3 } 10 x y d x d y$

Find the area of the part of hyperbolic paraboloid $z = y ^ { 2 } - x ^ { 2 }$ that lies between the cylinders $x ^ { 2 } + y ^ { 2 } = 1$ and $x ^ { 2 } + y ^ { 2 } = 9$ .

Find the mass of the lamina that occupies the region $D$ and has the given density function, if $D$ is bounded by the parabola $x = y ^ { 2 }$ and the line $y = x - 2$ . $\rho ( x , y ) = 3$

Calculate the iterated integral. $\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { y } 5 \cos \left( y ^ { 2 } \right) d x d y$

Find the area of the part of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 36 z$ that lies inside the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ . Select the correct answer.

Calculate the double integral. $\iint _ { R } \left( 9 x ^ { 2 } y ^ { 3 } - 15 x ^ { 4 } \right) d A , R = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq 4 \}$

Calculate the iterated integral. $\int _ { - 1 } ^ { 1 } \int _ { 0 } ^ { 1 } 5 y e ^ { x y } d x d y$

Use cylindrical or spherical coordinates, whichever seems more appropriate, to evaluate $\iiint _ { E } z d V$ where $E$ lies above the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ and below the plane $z = 25$ .

Sketch the solid bounded by the graphs of the equations $z = x ^ { 2 } + y ^ { 2 }$ and $z = 8 - x ^ { 2 } - y ^ { 2 }$ , and then use a triple integral to find the volume of the solid.

Use a triple integral to find the volume of the solid bounded by $x = y ^ { 2 }$ and the planes $z = 0$ and $x + z = 3$ . Select the correct answer.

Find the center of mass of the lamina that occupies the region $D$ and has the given density function, if $D$ is bounded by the parabola $y = 1 - x ^ { 2 }$ and the $x$ -axis. $\rho ( x , y ) = 4 y$

Evaluate the iterated integral. Select the correct answer. $\int _ { 1 } ^ { 3 } \int _ { y } ^ { 3 } 12 x y d x d y$