Question 1

Evaluate the iterated integral $\int _ { 0 } ^ { 2 } \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { x } 6 x y ^ { 2 } \sin z d z d y d x$ .

Accepted Answer

A)  $\sqrt { 2 }$ 
B) 2 
C) 4 
D) 16 
E) 12
F) $2 \sqrt { 2 }$
G)48
H)8 
A)  $\sqrt { 2 }$ 
B) 2 
C) 4 
D) 16 
E) 12
F) $2 \sqrt { 2 }$
G)48
H)8 H

Question 2

Use polar coordinates to combine the sum $\int _ { \sqrt { 2 } } ^ { 2 } \int _ { \sqrt { 4 - x ^ { 2 } } } ^ { x } x y d y d x + \int _ { 2 } ^ { 2 \sqrt { 2 } } \int _ { 0 } ^ { x } x y d y d x + \int _ { 2 \sqrt { 2 } } ^ { 4 } \int _ { 0 } ^ { \sqrt { 16 - x ^ { 2 } } } x y d y d x$ into one double integral. Then evaluate the integral.

Accepted Answer

$\int _ { 0 } ^ { x / 4 } \int _ { 2 } ^ { 4 } r ^ { 3 } \sin \theta \cos \theta d r d \theta = 15$

Question 3

Suppose the volume of a solid is given by $V = \int _ { 0 } ^ { 3 } \int _ { 0 } ^ { ( 3 - z ) / 2 } \int _ { 0 } ^ { 4 - x ^ { 2 } } d y d x d z$ .(a) Sketch the solid whose volume is given by V .   (b) Evaluate the integral to find the volume of the solid.

Accepted Answer

(a)   (b) $\frac { 261 } { 32 }$

Question 4

Find the x-coordinate of the center of mass of the lamina that occupies the part of the disk $x ^ { 2 } + y ^ { 2 } \leq 1$ in the first quadrant and has density function p(x, y) = xy.

Accepted Answer

A)  $\frac { 15 } { 8 }$ 
B)  $\frac { 1 } { 5 }$ 
C)  $\frac { 1 } { 3 }$ 
D)  $\frac { 1 } { 2 }$ 
E)  $\frac { 8 } { 15 }$
F) $\frac { 1 } { 8 }$

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

H)1 
A)  $\frac { 15 } { 8 }$ 
B)  $\frac { 1 } { 5 }$ 
C)  $\frac { 1 } { 3 }$ 
D)  $\frac { 1 } { 2 }$ 
E)  $\frac { 8 } { 15 }$
F) $\frac { 1 } { 8 }$

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

H)1

Question 5

Find the area of the surface cut from the cone $z = 1 - \sqrt { x ^ { 2 } + y ^ { 2 } }$ by the cylinder $x ^ { 2 } + y ^ { 2 } = y$ .

Accepted Answer

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

Question 6

Write $\iint _ { R } f ( x , y ) d A$ as an iterated integral in polar coordinates, where R is the region shown below.

Accepted Answer

The answer of Write \(\iint _ { R } f...

Question 7

Evaluate the iterated integral $\int _ { - 1 } ^ { 2 } \int _ { 0 } ^ { 1 - x } 4 - y d y d x$ .

Accepted Answer

A)  $\frac { 9 } { 4 }$ 
B)  $\frac { 3 } { 2 }$ 
C)  $- \frac { 9 } { 2 }$ 
D) 4 
E) 
$\frac { 14 } { 5 }$
F) $- \frac { 4 } { 5 }$
 
G) $\frac { 9 } { 2 }$
H)-4 
A)  $\frac { 9 } { 4 }$ 
B)  $\frac { 3 } { 2 }$ 
C)  $- \frac { 9 } { 2 }$ 
D) 4 
E) 
$\frac { 14 } { 5 }$
F) $- \frac { 4 } { 5 }$
 
G) $\frac { 9 } { 2 }$
H)-4

Question 8

Let $f ( x , y ) = x ^ { 2 }$ , and let $R = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq 1 \}$ . Let R be partitioned into two subrectangles by the line $x = \frac { 1 } { 2 }$ , and let $\left( x _ { i } ^ { * } , y _ { j } ^ { * } \right)$ be the center of Rij. Calculate the double Riemann sum of f.

Accepted Answer

A)  $\frac { 3 } { 16 }$ 
B)  $\frac { 1 } { 4 }$ 
C)  $\frac { 5 } { 16 }$ 
D)  $\frac { 3 } { 8 }$ 
E)  $\frac { 7 } { 16 }$
F) $\frac { 1 } { 2 }$

G) $\frac { 9 } { 16 }$ 
 
H) $\frac { 3 } { 4 }$ 
A)  $\frac { 3 } { 16 }$ 
B)  $\frac { 1 } { 4 }$ 
C)  $\frac { 5 } { 16 }$ 
D)  $\frac { 3 } { 8 }$ 
E)  $\frac { 7 } { 16 }$
F) $\frac { 1 } { 2 }$

G) $\frac { 9 } { 16 }$ 
 
H) $\frac { 3 } { 4 }$

Question 9

Rewrite the integral $\int _ { - 1 } ^ { 1 } \int _ { 0 } ^ { \sqrt { 1 - y ^ { 2 } } } x d x d y$ in terms of polar coordinates, then evaluate the integral.

Accepted Answer

The answer of Rewrite the integral \(\int _ { -...

Question 10

Evaluate $\int _ { 0 } ^ { 2 } \int _ { 0 } ^ { \sqrt { 4 - x ^ { 2 } } } \int _ { 0 } ^ { 4 - x ^ { 2 } - y ^ { 2 } } x ^ { 2 } d z d y d x$ by changing to cylindrical coordinates.

Accepted Answer

The answer of Evaluate \(\int _ { 0 } ^...

Question 11

Find the area of the part of the cone $4 x ^ { 2 } + 4 y ^ { 2 } = z ^ { 2 }$ that is above the region in the first quadrant bounded by the line y = x and the parabola $y = x ^ { 2 }$ .

Accepted Answer

A)  $\frac { 5 } { 6 }$ 
B)  $\frac { 5 } { 3 }$ 
C) $\frac { 2 } { 3 }$ 
D) 2 
E)  $\frac { \sqrt { 5 } } { 6 }$ 
F) $\frac { \sqrt { 5 } } { 3 }$ 
 
G) $\frac { \sqrt { 2 } } { 5 }$ 
H)6 
A)  $\frac { 5 } { 6 }$ 
B)  $\frac { 5 } { 3 }$ 
C) $\frac { 2 } { 3 }$ 
D) 2 
E)  $\frac { \sqrt { 5 } } { 6 }$ 
F) $\frac { \sqrt { 5 } } { 3 }$ 
 
G) $\frac { \sqrt { 2 } } { 5 }$ 
H)6

Question 12

Find the mass of the solid that occupies the region bounded by $x ^ { 2 } + y ^ { 2 } = 1$ , z = 2, and z = 0 and has density function $\rho  ( x , y , z ) = z$ .

Accepted Answer

A)  $\frac { \pi } { 4 }$ 
B)  $\frac { \pi } { 3 }$ 
C)  $\frac { \pi } { 2 }$ 
D)  $\frac { 2 \pi } { 3 }$ 
E)  $\frac { 3 \pi } { 4 }$

F)$\pi$

G) $\frac { 4 \pi } { 3 }$

H)2$\pi$ 
A)  $\frac { \pi } { 4 }$ 
B)  $\frac { \pi } { 3 }$ 
C)  $\frac { \pi } { 2 }$ 
D)  $\frac { 2 \pi } { 3 }$ 
E)  $\frac { 3 \pi } { 4 }$

F)$\pi$

G) $\frac { 4 \pi } { 3 }$

H)2$\pi$

Question 13

Let $f ( x , y ) = x y$ , and let $R = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq 1 \}$ . Let R be partitioned into four subrectangles by the lines $x = \frac { 1 } { 2 }$ and $y = \frac { 1 } { 2 }$ , and let $\left( x _ { i } ^ { * } , y _ { j } ^ { * } \right)$ be the upper right corner of Rij. Calculate the double Riemann sum of f.

Accepted Answer

A)  $\frac { 3 } { 16 }$ 
B)  $\frac { 1 } { 4 }$ 
C)  $\frac { 5 } { 16 }$ 
D)  $\frac { 3 } { 8 }$ 
E)  $\frac { 7 } { 16 }$ 
F) $\frac { 1 } { 2 }$

G) $\frac { 9 } { 16 }$ 
 
H) $\frac { 3 } { 4 }$ 
A)  $\frac { 3 } { 16 }$ 
B)  $\frac { 1 } { 4 }$ 
C)  $\frac { 5 } { 16 }$ 
D)  $\frac { 3 } { 8 }$ 
E)  $\frac { 7 } { 16 }$ 
F) $\frac { 1 } { 2 }$

G) $\frac { 9 } { 16 }$ 
 
H) $\frac { 3 } { 4 }$

Question 14

Rewrite $$\iint _ { R } f ( x , y ) d A$$ as an iterated integral with x as the variable of integration in the
outer integral, where R is the region shown below.

Accepted Answer

The answer of Rewrite \[\iint _ { R } f...

Question 15

Compute the Riemann sum for the double integral $\iint _ { R } 4 d A$ where $R = [ 0,6 ] \times [ 0,2 ]$ for the given grid and choice of sample points.

Accepted Answer

The answer of Compute the Riemann sum for the double...

Question 16

Evaluate the iterated integral $\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { x } \int _ { 0 } ^ { y ^ { 2 } } d x d y d z$ .

Accepted Answer

A)  $\frac { 1 } { 120 }$ 
B)  $\frac { 1 } { 90 }$ 
C)  $\frac { 1 } { 60 }$ 
D)  $\frac { 1 } { 48 }$ 
E)  $\frac { 1 } { 36 }$ 
F) $\frac { 1 } { 24 }$

G) $\frac { 1 } { 12 }$

H) $\frac { 1 } { 6 }$ 
A)  $\frac { 1 } { 120 }$ 
B)  $\frac { 1 } { 90 }$ 
C)  $\frac { 1 } { 60 }$ 
D)  $\frac { 1 } { 48 }$ 
E)  $\frac { 1 } { 36 }$ 
F) $\frac { 1 } { 24 }$

G) $\frac { 1 } { 12 }$

H) $\frac { 1 } { 6 }$

Question 17

Evaluate $\iint _ { D } \cos \left( x ^ { 2 } + y ^ { 2 } \right) d A$ , where $D = \left\{ ( x , y ) \mid x ^ { 2 } + y ^ { 2 } \leq 1 \right\}$ , the unit disk.

Accepted Answer

The answer of Evaluate \(\iint _ { D } \cos...

Question 18

Find the surface area of the surface z = xy inside the cylinder $x ^ { 2 } + y ^ { 2 } = 1$ .

Accepted Answer

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

Question 19

Evaluate the triple integral $\iiint _ { E } x d V$ , where $E = \{ ( x , y , z ) \mid 0 \leq x \leq y , 0 \leq y \leq 1,0 \leq z \leq 1 \}$ .

Accepted Answer

A)  $\frac { 1 } { 6 }$ 
B)  $\frac { 1 } { 4 }$ 
C)  $\frac { 1 } { 3 }$ 
D)  $\frac { 1 } { 2 }$ 
E)  $\frac { 2 } { 3 }$
F) $\frac { 1 } { 12 }$

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

H)1 
A)  $\frac { 1 } { 6 }$ 
B)  $\frac { 1 } { 4 }$ 
C)  $\frac { 1 } { 3 }$ 
D)  $\frac { 1 } { 2 }$ 
E)  $\frac { 2 } { 3 }$
F) $\frac { 1 } { 12 }$

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

H)1

Question 20

Find the Jacobian of the transformation x = 3u + v, y = u - 2w, z = v + w.

Accepted Answer

A) 1 
B) 6 
C) -5 
D) 4 
E) -1
F)-6
G)5
H)-4 
A) 1 
B) 6 
C) -5 
D) 4 
E) -1
F)-6
G)5
H)-4

Evaluate the iterated integral $\int _ { 0 } ^ { 2 } \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { x } 6 x y ^ { 2 } \sin z d z d y d x$ .

Find the x-coordinate of the center of mass of the lamina that occupies the part of the disk $x ^ { 2 } + y ^ { 2 } \leq 1$ in the first quadrant and has density function p(x, y) = xy.

Find the area of the surface cut from the cone $z = 1 - \sqrt { x ^ { 2 } + y ^ { 2 } }$ by the cylinder $x ^ { 2 } + y ^ { 2 } = y$ .

Write $\iint _ { R } f ( x , y ) d A$ as an iterated integral in polar coordinates, where R is the region shown below. $Write \iint _ { R } f ( x , y ) d A as an iterated integral in polar coordinates, where R is the region shown below.$

Evaluate the iterated integral $\int _ { - 1 } ^ { 2 } \int _ { 0 } ^ { 1 - x } 4 - y d y d x$ .

Rewrite the integral $\int _ { - 1 } ^ { 1 } \int _ { 0 } ^ { \sqrt { 1 - y ^ { 2 } } } x d x d y$ in terms of polar coordinates, then evaluate the integral.

Evaluate $\int _ { 0 } ^ { 2 } \int _ { 0 } ^ { \sqrt { 4 - x ^ { 2 } } } \int _ { 0 } ^ { 4 - x ^ { 2 } - y ^ { 2 } } x ^ { 2 } d z d y d x$ by changing to cylindrical coordinates.

Find the area of the part of the cone $4 x ^ { 2 } + 4 y ^ { 2 } = z ^ { 2 }$ that is above the region in the first quadrant bounded by the line y = x and the parabola $y = x ^ { 2 }$ .

Find the mass of the solid that occupies the region bounded by $x ^ { 2 } + y ^ { 2 } = 1$ , z = 2, and z = 0 and has density function $\rho ( x , y , z ) = z$ .

Evaluate the iterated integral $\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { x } \int _ { 0 } ^ { y ^ { 2 } } d x d y d z$ .

Evaluate $\iint _ { D } \cos \left( x ^ { 2 } + y ^ { 2 } \right) d A$ , where $D = \left\{ ( x , y ) \mid x ^ { 2 } + y ^ { 2 } \leq 1 \right\}$ , the unit disk.

Find the surface area of the surface z = xy inside the cylinder $x ^ { 2 } + y ^ { 2 } = 1$ .

Evaluate the triple integral $\iiint _ { E } x d V$ , where $E = \{ ( x , y , z ) \mid 0 \leq x \leq y , 0 \leq y \leq 1,0 \leq z \leq 1 \}$ .

Find the Jacobian of the transformation x = 3u + v, y = u - 2w, z = v + w.

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Exam 12: Multiple Integrals

Evaluate the iterated integral ∫02∫01∫0x6xy2sin⁡zdzdydx\int _ { 0 } ^ { 2 } \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { x } 6 x y ^ { 2 } \sin z d z d y d x∫02​∫01​∫0x​6xy2sinzdzdydx .

Find the x-coordinate of 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 and has density function p(x, y) = xy.

Find the area of the surface cut from the cone z=1−x2+y2z = 1 - \sqrt { x ^ { 2 } + y ^ { 2 } }z=1−x2+y2​ by the cylinder x2+y2=yx ^ { 2 } + y ^ { 2 } = yx2+y2=y .

Write ∬Rf(x,y)dA\iint _ { R } f ( x , y ) d A∬R​f(x,y)dA as an iterated integral in polar coordinates, where R is the region shown below.

Evaluate the iterated integral ∫−12∫01−x4−ydydx\int _ { - 1 } ^ { 2 } \int _ { 0 } ^ { 1 - x } 4 - y d y d x∫−12​∫01−x​4−ydydx .

Rewrite the integral ∫−11∫01−y2xdxdy\int _ { - 1 } ^ { 1 } \int _ { 0 } ^ { \sqrt { 1 - y ^ { 2 } } } x d x d y∫−11​∫01−y2​​xdxdy in terms of polar coordinates, then evaluate the integral.

Evaluate ∫02∫04−x2∫04−x2−y2x2dzdydx\int _ { 0 } ^ { 2 } \int _ { 0 } ^ { \sqrt { 4 - x ^ { 2 } } } \int _ { 0 } ^ { 4 - x ^ { 2 } - y ^ { 2 } } x ^ { 2 } d z d y d x∫02​∫04−x2​​∫04−x2−y2​x2dzdydx by changing to cylindrical coordinates.

Find the area of the part of the cone 4x2+4y2=z24 x ^ { 2 } + 4 y ^ { 2 } = z ^ { 2 }4x2+4y2=z2 that is above the region in the first quadrant bounded by the line y = x and the parabola y=x2y = x ^ { 2 }y=x2 .

Find the mass of the solid that occupies the region bounded by x2+y2=1x ^ { 2 } + y ^ { 2 } = 1x2+y2=1 , z = 2, and z = 0 and has density function ρ(x,y,z)=z\rho ( x , y , z ) = zρ(x,y,z)=z .

Rewrite ∬Rf(x,y)dA\iint _ { R } f ( x , y ) d A∬R​f(x,y)dA as an iterated integral with x as the variable of integration in the outer integral, where R is the region shown below.

Compute the Riemann sum for the double integral ∬R4dA\iint _ { R } 4 d A∬R​4dA where R=[0,6]×[0,2]R = [ 0,6 ] \times [ 0,2 ]R=[0,6]×[0,2] for the given grid and choice of sample points.

Evaluate the iterated integral ∫01∫0x∫0y2dxdydz\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { x } \int _ { 0 } ^ { y ^ { 2 } } d x d y d z∫01​∫0x​∫0y2​dxdydz .

Evaluate ∬Dcos⁡(x2+y2)dA\iint _ { D } \cos \left( x ^ { 2 } + y ^ { 2 } \right) d A∬D​cos(x2+y2)dA , where D={(x,y)∣x2+y2≤1}D = \left\{ ( x , y ) \mid x ^ { 2 } + y ^ { 2 } \leq 1 \right\}D={(x,y)∣x2+y2≤1} , the unit disk.

Find the surface area of the surface z = xy inside the cylinder x2+y2=1x ^ { 2 } + y ^ { 2 } = 1x2+y2=1 .

Evaluate the triple integral ∭ExdV\iiint _ { E } x d V∭E​xdV , where E={(x,y,z)∣0≤x≤y,0≤y≤1,0≤z≤1}E = \{ ( x , y , z ) \mid 0 \leq x \leq y , 0 \leq y \leq 1,0 \leq z \leq 1 \}E={(x,y,z)∣0≤x≤y,0≤y≤1,0≤z≤1} .

Find the Jacobian of the transformation x = 3u + v, y = u - 2w, z = v + w.

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Evaluate the iterated integral $\int _ { 0 } ^ { 2 } \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { x } 6 x y ^ { 2 } \sin z d z d y d x$ .

Find the x-coordinate of the center of mass of the lamina that occupies the part of the disk $x ^ { 2 } + y ^ { 2 } \leq 1$ in the first quadrant and has density function p(x, y) = xy.

Find the area of the surface cut from the cone $z = 1 - \sqrt { x ^ { 2 } + y ^ { 2 } }$ by the cylinder $x ^ { 2 } + y ^ { 2 } = y$ .

Write $\iint _ { R } f ( x , y ) d A$ as an iterated integral in polar coordinates, where R is the region shown below. $Write \iint _ { R } f ( x , y ) d A as an iterated integral in polar coordinates, where R is the region shown below.$

Evaluate the iterated integral $\int _ { - 1 } ^ { 2 } \int _ { 0 } ^ { 1 - x } 4 - y d y d x$ .

Rewrite the integral $\int _ { - 1 } ^ { 1 } \int _ { 0 } ^ { \sqrt { 1 - y ^ { 2 } } } x d x d y$ in terms of polar coordinates, then evaluate the integral.

Evaluate $\int _ { 0 } ^ { 2 } \int _ { 0 } ^ { \sqrt { 4 - x ^ { 2 } } } \int _ { 0 } ^ { 4 - x ^ { 2 } - y ^ { 2 } } x ^ { 2 } d z d y d x$ by changing to cylindrical coordinates.

Find the area of the part of the cone $4 x ^ { 2 } + 4 y ^ { 2 } = z ^ { 2 }$ that is above the region in the first quadrant bounded by the line y = x and the parabola $y = x ^ { 2 }$ .

Find the mass of the solid that occupies the region bounded by $x ^ { 2 } + y ^ { 2 } = 1$ , z = 2, and z = 0 and has density function $\rho ( x , y , z ) = z$ .

Evaluate the iterated integral $\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { x } \int _ { 0 } ^ { y ^ { 2 } } d x d y d z$ .

Evaluate $\iint _ { D } \cos \left( x ^ { 2 } + y ^ { 2 } \right) d A$ , where $D = \left\{ ( x , y ) \mid x ^ { 2 } + y ^ { 2 } \leq 1 \right\}$ , the unit disk.

Find the surface area of the surface z = xy inside the cylinder $x ^ { 2 } + y ^ { 2 } = 1$ .

Evaluate the triple integral $\iiint _ { E } x d V$ , where $E = \{ ( x , y , z ) \mid 0 \leq x \leq y , 0 \leq y \leq 1,0 \leq z \leq 1 \}$ .