Question 1

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 _ {...

Question 2

Evaluate the double integral by first identifying it as the volume of a solid. $$\iint _ { R } ( 15 - 2 x ) d A , R = \{ ( x , y ) \mid 2 \leq x \leq 5,3 \leq y \leq 8 \}$$

Accepted Answer

A)  300 
B)  -100 
C)  0 
D)  200 
E)  100 
A)  300 
B)  -100 
C)  0 
D)  200 
E)  100

Question 3

Use spherical coordinate to find the volume above the cone $z ^ { 2 } = x ^ { 2 } + y ^ { 2 }$ and inside sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 2 a z$.

Accepted Answer

The answer of Use spherical coordinate to find the volume...

Question 4

Use polar coordinates to find the volume of the solid under the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ and above the disk $x ^ { 2 } + y ^ { 2 } \leq 9$.

Accepted Answer

A)  $81 \pi$ 
B)  $162 \pi$ 
C)  $324 \pi$ 
D)  $27 \pi$ 
E)  $40.5 \pi$ 
A)  $81 \pi$ 
B)  $162 \pi$ 
C)  $324 \pi$ 
D)  $27 \pi$ 
E)  $40.5 \pi$

Question 5

Find the Jacobian of the transformation. $$x = 5 \alpha \sin \beta , y = 4 \alpha \cos \beta$$

Accepted Answer

A)  $\frac { \partial ( x , y ) } { \partial ( \alpha , \beta ) } = 9 \alpha$ 
B)  $\frac { \partial ( x , y ) } { \partial ( \alpha , \beta ) } = 36 \alpha$ 
C)  $\frac { \partial ( x , y ) } { \partial ( \alpha , \beta ) } = - 20 \alpha$ 
D)  $\frac { \partial ( x , y ) } { \partial ( \alpha , \beta ) } = - 20 \alpha \sin \beta \cos \beta$ 
E)  $\frac { \partial ( x , y ) } { \partial ( \alpha , \beta ) } = - \alpha$ 
A)  $\frac { \partial ( x , y ) } { \partial ( \alpha , \beta ) } = 9 \alpha$ 
B)  $\frac { \partial ( x , y ) } { \partial ( \alpha , \beta ) } = 36 \alpha$ 
C)  $\frac { \partial ( x , y ) } { \partial ( \alpha , \beta ) } = - 20 \alpha$ 
D)  $\frac { \partial ( x , y ) } { \partial ( \alpha , \beta ) } = - 20 \alpha \sin \beta \cos \beta$ 
E)  $\frac { \partial ( x , y ) } { \partial ( \alpha , \beta ) } = - \alpha$

Question 6

Calculate the iterated integral. $$\int _ { 1 } ^ { 4 } \int _ { 0 } ^ { 3 } ( 3 + 4 x y ) d x d y$$

Accepted Answer

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

Question 7

Use the given transformation to evaluate the integral.
$\iint _ { R } x y d A$, where $R$ is the region in the first quadrant bounded by the lines $y = x , y = 3 x$ and the hyperbolas $x y = 2 , x y = 4 ; x = \frac { u } { v } , y = v$.

Accepted Answer

A)  $9.447$ 
B)  $3.296$ 
C)  $8.841$ 
D)  $4.447$ 
E)  $5.088$ 
A)  $9.447$ 
B)  $3.296$ 
C)  $8.841$ 
D)  $4.447$ 
E)  $5.088$

Question 8

Find the area of the surface.
The part of the surface $z = 25 - x ^ { 2 } - y ^ { 2 }$ that lies above the $x y$-plane.

Accepted Answer

A)  $( \sqrt { 626 } - 626 )$ 
B)  $\frac { 2 } { 75 } ( 626 \sqrt { 626 } + 1 )$ 
C)  $\frac { 2 } { 75 } \pi ( 626 \sqrt { 626 } - 1 )$ 
D)  $\frac { 2 } { 75 } ( \sqrt { 626 } + 1 )$ 
E)  $\pi ( 626 \sqrt { 626 } - 1 )$ 
A)  $( \sqrt { 626 } - 626 )$ 
B)  $\frac { 2 } { 75 } ( 626 \sqrt { 626 } + 1 )$ 
C)  $\frac { 2 } { 75 } \pi ( 626 \sqrt { 626 } - 1 )$ 
D)  $\frac { 2 } { 75 } ( \sqrt { 626 } + 1 )$ 
E)  $\pi ( 626 \sqrt { 626 } - 1 )$

Question 9

Evaluate the integral by changing to polar coordinates. $\iint _ { D } e ^ { - x ^ { 2 } - y ^ { 2 } } d A$
$D$ is the region bounded by the semicircle $x = \sqrt { 9 - y ^ { 2 } }$ and the $y$-axis.

Accepted Answer

The answer of Evaluate the integral by changing to polar...

Question 10

Evaluate the double integral by first identifying it as the volume of a solid. $$\iint _ { R } ( 15 - 2 x ) d A , R = \{ ( x , y ) \mid 3 \leq x \leq 7,3 \leq y \leq 5 \}$$

Accepted Answer

A)  40 
B)  $- 160$ 
C)  $- 60$ 
D)  140 
E)  240 
A)  40 
B)  $- 160$ 
C)  $- 60$ 
D)  140 
E)  240

Question 11

Calculate the double integral. $$\iint _ { R } \left( 6 x ^ { 2 } y ^ { 3 } - 10 x ^ { 4 } \right) d A , R = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq 4 \}$$

Accepted Answer

A)  190 
B)  170 
C)  130 
D)  150 
E)  120 
A)  190 
B)  170 
C)  130 
D)  150 
E)  120

Question 12

Use spherical coordinates.
Evaluate $\iiint _ { B } \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { 2 } d V$, where $B$ is the ball with center the origin and radius 4 .

Accepted Answer

A)  $\frac { 65536 } { 7 } \pi$ 
B)  $\frac { 4374 } { 7 }$ 
C)  $\frac { 4374 } { 7 } \pi$ 
D)  $\frac { 559872 } { 7 } \pi$ 
E)  None of these 
A)  $\frac { 65536 } { 7 } \pi$ 
B)  $\frac { 4374 } { 7 }$ 
C)  $\frac { 4374 } { 7 } \pi$ 
D)  $\frac { 559872 } { 7 } \pi$ 
E)  None of these

Question 13

Sketch the solid bounded by the graphs of the equations $z = x ^ { 2 } + y ^ { 2 }$ and $z = 50 - x ^ { 2 } - y ^ { 2 }$, 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 14

Calculate the iterated integral.Round your answer to two decimal places. $$\int _ { 0 } ^ { 6 } \int _ { 0 } ^ { 3 } \sqrt { 4 x + 4 y } d x d y$$

Accepted Answer

A)  $94.26$ 
B)  $134.26$ 
C)  $114.26$ 
D)  $84.26$ 
E)  $74.26$ 
A)  $94.26$ 
B)  $134.26$ 
C)  $114.26$ 
D)  $84.26$ 
E)  $74.26$

Question 15

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

Accepted Answer

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

Question 16

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

A)  250 
B)  230 
C)  190 
D)  180 
E)  210 
A)  250 
B)  230 
C)  190 
D)  180 
E)  210

Question 17

Use cylindrical coordinates to evaluate $\iiint _ { T } \sqrt { x ^ { 2 } + y ^ { 2 } } d V$, where $T$ is the solid bounded by the cylinder $x ^ { 2 } + y ^ { 2 } = 1$ and the planes $z = 4$ and $z = 8$.

Accepted Answer

A)  $\frac { 8 } { 3 } \pi$ 
B)  $4 \pi$ 
C)  $32 \pi$ 
D)  $48 \pi$ 
A)  $\frac { 8 } { 3 } \pi$ 
B)  $4 \pi$ 
C)  $32 \pi$ 
D)  $48 \pi$

Question 18

Find the center of mass of the lamina of the region shown if the density of the circular lamina is four times that of the rectangular lamina.

Accepted Answer

The answer of Find the center of mass of the...

Question 19

Use the Midpoint Rule with four squares of equal size to estimate the double integral. $$\iint _ { R } \cos \left( x ^ { 4 } + y ^ { 4 } \right) d A , R = \{ ( x , y ) \mid 0 \leq x \leq 0.5,0 \leq y \leq 0.5 \}$$

Accepted Answer

A)  $0.575$ 
B)  $0.6622$ 
C)  $0.2499$ 
D)  $0.4721$ 
E)  $0.7622$ 
A)  $0.575$ 
B)  $0.6622$ 
C)  $0.2499$ 
D)  $0.4721$ 
E)  $0.7622$

Question 20

Find the center of mass of a lamina in the shape of an isosceles right triangle with equal sides of length $a = 25$ if the density at any point is proportional to the square of the distance from the vertex opposite the hypotenuse. Assume the vertex opposite the hypotenuse is located at $( 0,0 )$, and that the sides are along the positive axes.

Accepted Answer

A)  $( 10,10 )$ 
B)  $( 10,25 )$ 
C)  $( 25,25 )$ 
D)  $( 5,10 )$ 
E)  None of these 
A)  $( 10,10 )$ 
B)  $( 10,25 )$ 
C)  $( 25,25 )$ 
D)  $( 5,10 )$ 
E)  None of these

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

Evaluate the double integral by first identifying it as the volume of a solid. $\iint _ { R } ( 15 - 2 x ) d A , R = \{ ( x , y ) \mid 2 \leq x \leq 5,3 \leq y \leq 8 \}$

Use spherical coordinate to find the volume above the cone $z ^ { 2 } = x ^ { 2 } + y ^ { 2 }$ and inside sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 2 a z$ .

Use polar coordinates to find the volume of the solid under the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ and above the disk $x ^ { 2 } + y ^ { 2 } \leq 9$ .

Find the Jacobian of the transformation. $x = 5 \alpha \sin \beta , y = 4 \alpha \cos \beta$

Calculate the iterated integral. $\int _ { 1 } ^ { 4 } \int _ { 0 } ^ { 3 } ( 3 + 4 x y ) d x d y$

Use the given transformation to evaluate the integral. $\iint _ { R } x y d A$ , where $R$ is the region in the first quadrant bounded by the lines $y = x , y = 3 x$ and the hyperbolas $x y = 2 , x y = 4 ; x = \frac { u } { v } , y = v$ .

Find the area of the surface. The part of the surface $z = 25 - x ^ { 2 } - y ^ { 2 }$ that lies above the $x y$ -plane.

Evaluate the integral by changing to polar coordinates. $\iint _ { D } e ^ { - x ^ { 2 } - y ^ { 2 } } d A$ $D$ is the region bounded by the semicircle $x = \sqrt { 9 - y ^ { 2 } }$ and the $y$ -axis.

Evaluate the double integral by first identifying it as the volume of a solid. $\iint _ { R } ( 15 - 2 x ) d A , R = \{ ( x , y ) \mid 3 \leq x \leq 7,3 \leq y \leq 5 \}$

Calculate the double integral. $\iint _ { R } \left( 6 x ^ { 2 } y ^ { 3 } - 10 x ^ { 4 } \right) d A , R = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq 4 \}$

Use spherical coordinates. Evaluate $\iiint _ { B } \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { 2 } d V$ , where $B$ is the ball with center the origin and radius 4 .

Sketch the solid bounded by the graphs of the equations $z = x ^ { 2 } + y ^ { 2 }$ and $z = 50 - x ^ { 2 } - y ^ { 2 }$ , and then use a triple integral to find the volume of the solid.

Calculate the iterated integral.Round your answer to two decimal places. $\int _ { 0 } ^ { 6 } \int _ { 0 } ^ { 3 } \sqrt { 4 x + 4 y } d x d y$

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

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

Use cylindrical coordinates to evaluate $\iiint _ { T } \sqrt { x ^ { 2 } + y ^ { 2 } } d V$ , where $T$ is the solid bounded by the cylinder $x ^ { 2 } + y ^ { 2 } = 1$ and the planes $z = 4$ and $z = 8$ .

Find the center of mass of the lamina of the region shown if the density of the circular lamina is four times that of the rectangular lamina.

Use the Midpoint Rule with four squares of equal size to estimate the double integral. $\iint _ { R } \cos \left( x ^ { 4 } + y ^ { 4 } \right) d A , R = \{ ( x , y ) \mid 0 \leq x \leq 0.5,0 \leq y \leq 0.5 \}$

Functions and Models

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Techniques of Integration

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Exam 15: Multiple Integrals

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

Evaluate the double integral by first identifying it as the volume of a solid. ∬R(15−2x)dA,R={(x,y)∣2≤x≤5,3≤y≤8}\iint _ { R } ( 15 - 2 x ) d A , R = \{ ( x , y ) \mid 2 \leq x \leq 5,3 \leq y \leq 8 \}∬R​(15−2x)dA,R={(x,y)∣2≤x≤5,3≤y≤8}

Use spherical coordinate to find the volume above the cone z2=x2+y2z ^ { 2 } = x ^ { 2 } + y ^ { 2 }z2=x2+y2 and inside sphere x2+y2+z2=2azx ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 2 a zx2+y2+z2=2az .

Use polar coordinates to find the volume of the solid under the paraboloid z=x2+y2z = x ^ { 2 } + y ^ { 2 }z=x2+y2 and above the disk x2+y2≤9x ^ { 2 } + y ^ { 2 } \leq 9x2+y2≤9 .

Find the Jacobian of the transformation. x=5αsin⁡β,y=4αcos⁡βx = 5 \alpha \sin \beta , y = 4 \alpha \cos \betax=5αsinβ,y=4αcosβ

Calculate the iterated integral. ∫14∫03(3+4xy)dxdy\int _ { 1 } ^ { 4 } \int _ { 0 } ^ { 3 } ( 3 + 4 x y ) d x d y∫14​∫03​(3+4xy)dxdy

Find the area of the surface. The part of the surface z=25−x2−y2z = 25 - x ^ { 2 } - y ^ { 2 }z=25−x2−y2 that lies above the xyx yxy -plane.

Evaluate the integral by changing to polar coordinates. ∬De−x2−y2dA\iint _ { D } e ^ { - x ^ { 2 } - y ^ { 2 } } d A∬D​e−x2−y2dA DDD is the region bounded by the semicircle x=9−y2x = \sqrt { 9 - y ^ { 2 } }x=9−y2​ and the yyy -axis.

Evaluate the double integral by first identifying it as the volume of a solid. ∬R(15−2x)dA,R={(x,y)∣3≤x≤7,3≤y≤5}\iint _ { R } ( 15 - 2 x ) d A , R = \{ ( x , y ) \mid 3 \leq x \leq 7,3 \leq y \leq 5 \}∬R​(15−2x)dA,R={(x,y)∣3≤x≤7,3≤y≤5}

Calculate the double integral. ∬R(6x2y3−10x4)dA,R={(x,y)∣0≤x≤1,0≤y≤4}\iint _ { R } \left( 6 x ^ { 2 } y ^ { 3 } - 10 x ^ { 4 } \right) d A , R = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq 4 \}∬R​(6x2y3−10x4)dA,R={(x,y)∣0≤x≤1,0≤y≤4}

Use spherical coordinates. Evaluate ∭B(x2+y2+z2)2dV\iiint _ { B } \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { 2 } d V∭B​(x2+y2+z2)2dV , where BBB is the ball with center the origin and radius 4 .

Sketch the solid bounded by the graphs of the equations z=x2+y2z = x ^ { 2 } + y ^ { 2 }z=x2+y2 and z=50−x2−y2z = 50 - x ^ { 2 } - y ^ { 2 }z=50−x2−y2 , and then use a triple integral to find the volume of the solid.

Calculate the iterated integral.Round your answer to two decimal places. ∫06∫034x+4ydxdy\int _ { 0 } ^ { 6 } \int _ { 0 } ^ { 3 } \sqrt { 4 x + 4 y } d x d y∫06​∫03​4x+4y​dxdy

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}

Use cylindrical coordinates to evaluate ∭Tx2+y2dV\iiint _ { T } \sqrt { x ^ { 2 } + y ^ { 2 } } d V∭T​x2+y2​dV , where TTT is the solid bounded by the cylinder x2+y2=1x ^ { 2 } + y ^ { 2 } = 1x2+y2=1 and the planes z=4z = 4z=4 and z=8z = 8z=8 .

Find the center of mass of the lamina of the region shown if the density of the circular lamina is four times that of the rectangular lamina.

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Vector Calculus

Second-Order Differential Equations

Filters

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

Evaluate the double integral by first identifying it as the volume of a solid. $\iint _ { R } ( 15 - 2 x ) d A , R = \{ ( x , y ) \mid 2 \leq x \leq 5,3 \leq y \leq 8 \}$

Use spherical coordinate to find the volume above the cone $z ^ { 2 } = x ^ { 2 } + y ^ { 2 }$ and inside sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 2 a z$ .

Use polar coordinates to find the volume of the solid under the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ and above the disk $x ^ { 2 } + y ^ { 2 } \leq 9$ .

Find the Jacobian of the transformation. $x = 5 \alpha \sin \beta , y = 4 \alpha \cos \beta$

Calculate the iterated integral. $\int _ { 1 } ^ { 4 } \int _ { 0 } ^ { 3 } ( 3 + 4 x y ) d x d y$

Find the area of the surface. The part of the surface $z = 25 - x ^ { 2 } - y ^ { 2 }$ that lies above the $x y$ -plane.

Evaluate the integral by changing to polar coordinates. $\iint _ { D } e ^ { - x ^ { 2 } - y ^ { 2 } } d A$ $D$ is the region bounded by the semicircle $x = \sqrt { 9 - y ^ { 2 } }$ and the $y$ -axis.

Evaluate the double integral by first identifying it as the volume of a solid. $\iint _ { R } ( 15 - 2 x ) d A , R = \{ ( x , y ) \mid 3 \leq x \leq 7,3 \leq y \leq 5 \}$

Calculate the double integral. $\iint _ { R } \left( 6 x ^ { 2 } y ^ { 3 } - 10 x ^ { 4 } \right) d A , R = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq 4 \}$

Use spherical coordinates. Evaluate $\iiint _ { B } \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { 2 } d V$ , where $B$ is the ball with center the origin and radius 4 .

Sketch the solid bounded by the graphs of the equations $z = x ^ { 2 } + y ^ { 2 }$ and $z = 50 - x ^ { 2 } - y ^ { 2 }$ , and then use a triple integral to find the volume of the solid.

Calculate the iterated integral.Round your answer to two decimal places. $\int _ { 0 } ^ { 6 } \int _ { 0 } ^ { 3 } \sqrt { 4 x + 4 y } d x d y$

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

Use cylindrical coordinates to evaluate $\iiint _ { T } \sqrt { x ^ { 2 } + y ^ { 2 } } d V$ , where $T$ is the solid bounded by the cylinder $x ^ { 2 } + y ^ { 2 } = 1$ and the planes $z = 4$ and $z = 8$ .