Question 1

Evaluate $\int _ { 0 } ^ { 9 } \int _ { 0 } ^ { 1 } \sqrt { x y } d y d x$ .

Accepted Answer

A) 18 
B) 9 
C) 3 
D) 1 
E) 6
F)12
G)27
H)36 
A) 18 
B) 9 
C) 3 
D) 1 
E) 6
F)12
G)27
H)36

Question 2

Find the area of that part of the surface $z = x + y ^ { 2 }$ that lies above the triangle with vertices (0, 0), (1, 1), and (0, 1).

Accepted Answer

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

Question 3

A sphere of radius k has a volume of $\frac { 4 } { 3 } \pi k ^ { 3 }$ . Set up the iterated integrals in rectangular, cylindrical, and spherical coordinates to show this.

Accepted Answer

Rectangular coordina

Question 4

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

Accepted Answer

A)  $\frac { 1 } { 2 }$ 
B)  $\frac { 1 } { 3 }$ 
C) 
$\frac { 1 } { 4 }$ 
D)  $\frac { 3 } { 4 }$ 
E)  $\frac { 2 } { 3 }$ 
F) $\frac { 1 } { 6 }$

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

H)1 
A)  $\frac { 1 } { 2 }$ 
B)  $\frac { 1 } { 3 }$ 
C) 
$\frac { 1 } { 4 }$ 
D)  $\frac { 3 } { 4 }$ 
E)  $\frac { 2 } { 3 }$ 
F) $\frac { 1 } { 6 }$

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

H)1

Question 5

Calculate the double integral $\iint _ { R } \frac { 1 + x } { 1 + y } d A$ , where $R = \{ ( x , y ) \mid - 1 \leq x \leq 2,0 \leq y \leq 1 \}$ .

Accepted Answer

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

Question 6

Find the center of mass of the lamina that occupies the part of the disk $x ^ { 2 } + y ^ { 2 } \leq 1$ in the first quadrant if the density at any point is proportional to the square of its distance from the origin.

Accepted Answer

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

Question 7

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

Accepted Answer

The answer of Evaluate \(\iint _ { R } \sqrt...

Question 8

Find the area of the surface with vector equation r $( s , t ) = \langle s \cos t , s \sin t , s \rangle$ , $1 \leq s \leq 5$ , $0 \leq t \leq 2 \pi$ .

Accepted Answer

The answer of Find the area of the surface with...

Question 9

Use the method of iterated integration in order to evaluate the triple integral $\iiint _ { N } x d V$ where N is the region cut o from the first octant by the plane defined by x + y + z = 3.

Accepted Answer

The answer of Use the method of iterated integration in...

Question 10

Calculate the iterated integral $\int _ { 0 } ^ { \pi / 2 } \int _ { 0 } ^ { \pi / 2 } \sin ( x + y ) d y d x$ .

Accepted Answer

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

Question 11

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.

Accepted Answer

@#LAT-DLM& , image o

Question 12

Let R be the region bounded by $y = x ^ { 2 }$ , $y = 0$ , and $x = 1$ . Find the center of mass of a lamina in the shape of R with density function $p ( x , y ) = x y$ .

Accepted Answer

The answer of Let R be the region bounded by...

Question 13

The joint density function for a pair of random variables X and Y is $f ( x , y ) = \left\{ \begin{array} { l l } 
C ( x + 1 ) & \text { if } 0 \leq x \leq 1 \text { and } 0 \leq y \leq 1 \
0 & \text { otherwise }
\end{array} \right.$ (a) Find the value of C.(b) Find $P ( X + Y \leq 1 )$ .(c) Find $P ( X \leq 2 Y \leq 3 X )$ .

Accepted Answer

The answer of The joint density function for a pair...

Question 14

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

Accepted Answer

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) $\frac { 8 } { 9 }$ 
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 15

Find the volume, using triple integrals, of the region in the first octant beneath the plane x + 2y + 3z = 6.

Accepted Answer

The answer of Find the volume, using triple integrals, of...

Question 16

Find the mass and center of mass of the lamina that occupies the triangular region with vertices (0, 0), (1, 1), and (4, 0), and has density function $p ( x , y ) = x$ .

Accepted Answer

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

Question 17

Evaluate $\int _ { - 1 } ^ { 0 } \int _ { - \sqrt { 1 - y ^ { 2 } } } ^ { 0 } \cos \left( x ^ { 2 } + y ^ { 2 } \right) d x d y$ .

Accepted Answer

The answer of Evaluate \(\int _ { - 1 }...

Question 18

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

Accepted Answer

A)  $\frac { 1 } { 2 } ( \cos 1 - 1 )$ 
B)  $\frac { 1 } { 2 } ( 1 - \cos 1 )$ 
C)  $\frac { 1 } { 2 } \cos 1$ 
D) 1 
E)  $\frac { 1 } { 2 } ( \sin 1 - 1 )$ 
F) $\frac { 1 } { 2 } ( 1 - \sin 1 )$

G) $\frac { 1 } { 2 } \sin 1$ 
H)0 
A)  $\frac { 1 } { 2 } ( \cos 1 - 1 )$ 
B)  $\frac { 1 } { 2 } ( 1 - \cos 1 )$ 
C)  $\frac { 1 } { 2 } \cos 1$ 
D) 1 
E)  $\frac { 1 } { 2 } ( \sin 1 - 1 )$ 
F) $\frac { 1 } { 2 } ( 1 - \sin 1 )$

G) $\frac { 1 } { 2 } \sin 1$ 
H)0

Question 19

Evaluate the double integral $\iint _ { R } y ^ { 3 } \sin x \cos x d A$ , where $R = \left\{ ( x , y ) \mid 0 \leq x \leq \frac { \pi } { 2 } , 0 \leq y \leq 1 \right\}$ .

Accepted Answer

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

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

H)0 
A)  $\frac { 1 } { 4 }$ 
B)  $\frac { 1 } { 2 }$ 
C)  $\frac { \pi } { 4 }$ 
D)  $\frac { 1 } { 8 }$ 
E)  $\frac { 5 } { 4 }$ 
F) $\frac { 3 } { 2 }$

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

H)0

Question 20

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

Accepted Answer

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

Evaluate $\int _ { 0 } ^ { 9 } \int _ { 0 } ^ { 1 } \sqrt { x y } d y d x$ .

Find the area of that part of the surface $z = x + y ^ { 2 }$ that lies above the triangle with vertices (0, 0), (1, 1), and (0, 1).

A sphere of radius k has a volume of $\frac { 4 } { 3 } \pi k ^ { 3 }$ . Set up the iterated integrals in rectangular, cylindrical, and spherical coordinates to show this.

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

Calculate the double integral $\iint _ { R } \frac { 1 + x } { 1 + y } d A$ , where $R = \{ ( x , y ) \mid - 1 \leq x \leq 2,0 \leq y \leq 1 \}$ .

Find the center of mass of the lamina that occupies the part of the disk $x ^ { 2 } + y ^ { 2 } \leq 1$ in the first quadrant if the density at any point is proportional to the square of its distance from the origin.

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

Find the area of the surface with vector equation r $( s , t ) = \langle s \cos t , s \sin t , s \rangle$ , $1 \leq s \leq 5$ , $0 \leq t \leq 2 \pi$ .

Use the method of iterated integration in order to evaluate the triple integral $\iiint _ { N } x d V$ where N is the region cut o from the first octant by the plane defined by x + y + z = 3.

Calculate the iterated integral $\int _ { 0 } ^ { \pi / 2 } \int _ { 0 } ^ { \pi / 2 } \sin ( x + y ) d y d x$ .

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.

Let R be the region bounded by $y = x ^ { 2 }$ , $y = 0$ , and $x = 1$ . Find the center of mass of a lamina in the shape of R with density function $p ( x , y ) = x y$ .

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

Find the volume, using triple integrals, of the region in the first octant beneath the plane x + 2y + 3z = 6.

Find the mass and center of mass of the lamina that occupies the triangular region with vertices (0, 0), (1, 1), and (4, 0), and has density function $p ( x , y ) = x$ .

Evaluate $\int _ { - 1 } ^ { 0 } \int _ { - \sqrt { 1 - y ^ { 2 } } } ^ { 0 } \cos \left( x ^ { 2 } + y ^ { 2 } \right) d x d y$ .

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

Evaluate the double integral $\iint _ { R } y ^ { 3 } \sin x \cos x d A$ , where $R = \left\{ ( x , y ) \mid 0 \leq x \leq \frac { \pi } { 2 } , 0 \leq y \leq 1 \right\}$ .

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

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Exam 12: Multiple Integrals

Evaluate ∫09∫01xydydx\int _ { 0 } ^ { 9 } \int _ { 0 } ^ { 1 } \sqrt { x y } d y d x∫09​∫01​xy​dydx .

Find the area of that part of the surface z=x+y2z = x + y ^ { 2 }z=x+y2 that lies above the triangle with vertices (0, 0), (1, 1), and (0, 1).

A sphere of radius k has a volume of 43πk3\frac { 4 } { 3 } \pi k ^ { 3 }34​πk3 . Set up the iterated integrals in rectangular, cylindrical, and spherical coordinates to show this.

Evaluate the iterated integral ∫01∫x1ydydx\int _ { 0 } ^ { 1 } \int _ { x } ^ { 1 } y d y d x∫01​∫x1​ydydx .

Calculate the double integral ∬R1+x1+ydA\iint _ { R } \frac { 1 + x } { 1 + y } d A∬R​1+y1+x​dA , where R={(x,y)∣−1≤x≤2,0≤y≤1}R = \{ ( x , y ) \mid - 1 \leq x \leq 2,0 \leq y \leq 1 \}R={(x,y)∣−1≤x≤2,0≤y≤1} .

Find the center of mass of the lamina that occupies the part of the disk x2+y2≤1x ^ { 2 } + y ^ { 2 } \leq 1x2+y2≤1 in the first quadrant if the density at any point is proportional to the square of its distance from the origin.

Evaluate ∬Rb2x2+a2y2dA\iint _ { R } \sqrt { b ^ { 2 } x ^ { 2 } + a ^ { 2 } y ^ { 2 } } d A∬R​b2x2+a2y2​dA , where R is the region enclosed by the ellipse x2a2+y2b2=1\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1a2x2​+b2y2​=1 .

Find the area of the surface with vector equation r (s,t)=⟨scos⁡t,ssin⁡t,s⟩( s , t ) = \langle s \cos t , s \sin t , s \rangle(s,t)=⟨scost,ssint,s⟩ , 1≤s≤51 \leq s \leq 51≤s≤5 , 0≤t≤2π0 \leq t \leq 2 \pi0≤t≤2π .

Use the method of iterated integration in order to evaluate the triple integral ∭NxdV\iiint _ { N } x d V∭N​xdV where N is the region cut o from the first octant by the plane defined by x + y + z = 3.

Calculate the iterated integral ∫0π/2∫0π/2sin⁡(x+y)dydx\int _ { 0 } ^ { \pi / 2 } \int _ { 0 } ^ { \pi / 2 } \sin ( x + y ) d y d x∫0π/2​∫0π/2​sin(x+y)dydx .

Let R be the region bounded by y=x2y = x ^ { 2 }y=x2 , y=0y = 0y=0 , and x=1x = 1x=1 . Find the center of mass of a lamina in the shape of R with density function p(x,y)=xyp ( x , y ) = x yp(x,y)=xy .

Find the area of the region whose image under the transformation x = u + v, y = v - 2u is D={(x,y)∣−1≤x≤1,0≤y≤1−x2}D = \left\{ ( x , y ) \mid - 1 \leq x \leq 1,0 \leq y \leq 1 - x ^ { 2 } \right\}D={(x,y)∣−1≤x≤1,0≤y≤1−x2} .

Find the volume, using triple integrals, of the region in the first octant beneath the plane x + 2y + 3z = 6.

Find the mass and center of mass of the lamina that occupies the triangular region with vertices (0, 0), (1, 1), and (4, 0), and has density function p(x,y)=xp ( x , y ) = xp(x,y)=x .

Evaluate ∫−10∫−1−y20cos⁡(x2+y2)dxdy\int _ { - 1 } ^ { 0 } \int _ { - \sqrt { 1 - y ^ { 2 } } } ^ { 0 } \cos \left( x ^ { 2 } + y ^ { 2 } \right) d x d y∫−10​∫−1−y2​0​cos(x2+y2)dxdy .

Evaluate the iterated integral ∫01∫y1sin⁡(x2)dxdy\int _ { 0 } ^ { 1 } \int _ { y } ^ { 1 } \sin \left( x ^ { 2 } \right) d x d y∫01​∫y1​sin(x2)dxdy .

Evaluate the double integral ∬Ry3sin⁡xcos⁡xdA\iint _ { R } y ^ { 3 } \sin x \cos x d A∬R​y3sinxcosxdA , where R={(x,y)∣0≤x≤π2,0≤y≤1}R = \left\{ ( x , y ) \mid 0 \leq x \leq \frac { \pi } { 2 } , 0 \leq y \leq 1 \right\}R={(x,y)∣0≤x≤2π​,0≤y≤1} .

Evaluate the iterated integral ∫0x∫0ysin⁡xdxdy\int _ { 0 } ^ { x } \int _ { 0 } ^ { y } \sin x d x d y∫0x​∫0y​sinxdxdy .

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Evaluate $\int _ { 0 } ^ { 9 } \int _ { 0 } ^ { 1 } \sqrt { x y } d y d x$ .

Find the area of that part of the surface $z = x + y ^ { 2 }$ that lies above the triangle with vertices (0, 0), (1, 1), and (0, 1).

A sphere of radius k has a volume of $\frac { 4 } { 3 } \pi k ^ { 3 }$ . Set up the iterated integrals in rectangular, cylindrical, and spherical coordinates to show this.

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

Calculate the double integral $\iint _ { R } \frac { 1 + x } { 1 + y } d A$ , where $R = \{ ( x , y ) \mid - 1 \leq x \leq 2,0 \leq y \leq 1 \}$ .

Find the center of mass of the lamina that occupies the part of the disk $x ^ { 2 } + y ^ { 2 } \leq 1$ in the first quadrant if the density at any point is proportional to the square of its distance from the origin.

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

Find the area of the surface with vector equation r $( s , t ) = \langle s \cos t , s \sin t , s \rangle$ , $1 \leq s \leq 5$ , $0 \leq t \leq 2 \pi$ .

Use the method of iterated integration in order to evaluate the triple integral $\iiint _ { N } x d V$ where N is the region cut o from the first octant by the plane defined by x + y + z = 3.

Calculate the iterated integral $\int _ { 0 } ^ { \pi / 2 } \int _ { 0 } ^ { \pi / 2 } \sin ( x + y ) d y d x$ .

Let R be the region bounded by $y = x ^ { 2 }$ , $y = 0$ , and $x = 1$ . Find the center of mass of a lamina in the shape of R with density function $p ( x , y ) = x y$ .

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

Find the mass and center of mass of the lamina that occupies the triangular region with vertices (0, 0), (1, 1), and (4, 0), and has density function $p ( x , y ) = x$ .

Evaluate $\int _ { - 1 } ^ { 0 } \int _ { - \sqrt { 1 - y ^ { 2 } } } ^ { 0 } \cos \left( x ^ { 2 } + y ^ { 2 } \right) d x d y$ .

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

Evaluate the double integral $\iint _ { R } y ^ { 3 } \sin x \cos x d A$ , where $R = \left\{ ( x , y ) \mid 0 \leq x \leq \frac { \pi } { 2 } , 0 \leq y \leq 1 \right\}$ .

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