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


A) 94\frac { 9 } { 4 }

B) 32\frac { 3 } { 2 }
C) −92- \frac { 9 } { 2 }
D) 4
E)
145\frac { 14 } { 5 }
F) −45- \frac { 4 } { 5 }

G) 92\frac { 9 } { 2 }
H) -4

I) A) and E)
J) G) and H)

Correct Answer

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Suppose the volume of a solid is given by V=∫03∫0(3−z)/2∫04−x2dydxdzV = \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 . 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. (b) Evaluate the integral to find the volume of the solid.

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(a) 11eaa8e2_0fe6_2e27_96ab_734a30c83b2f_TB2033_00 (b) \(\frac { 261 } { 32 }\)

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

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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 .


A) 2\sqrt { 2 }
B) 2
C) 4
D) 16
E) 12
F) 222 \sqrt { 2 }
G) 48
H) 8

I) B) and E)
J) A) and D)

Correct Answer

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H

Use polar coordinates to combine the sum ∫22∫4−x2xxydydx+∫222∫0xxydydx+∫224∫016−x2xydydx\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.

Correct Answer

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

Evaluate the triple integral ∭ExdV\iiint _ { E } x d V , 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 \} .


A) 16\frac { 1 } { 6 }

B) 14\frac { 1 } { 4 }
C) 13\frac { 1 } { 3 }
D) 12\frac { 1 } { 2 }
E) 23\frac { 2 } { 3 }
F) 112\frac { 1 } { 12 }

G) 56\frac { 5 } { 6 }

H) 1

I) A) and C)
J) A) and F)

Correct Answer

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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 1 in the first quadrant and has density function p(x, y) = xy.


A) 158\frac { 15 } { 8 }

B) 15\frac { 1 } { 5 }
C) 13\frac { 1 } { 3 }
D) 12\frac { 1 } { 2 }
E) 815\frac { 8 } { 15 }
F) 18\frac { 1 } { 8 }

G) 56\frac { 5 } { 6 }

H) 1

I) A) and G)
J) E) and H)

Correct Answer

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

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


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

I) C) and F)
J) A) and B)

Correct Answer

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Let f(x,y) =x2f ( x , y ) = x ^ { 2 } , and let R={(x,y) ∣0≤x≤1,0≤y≤1}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=12x = \frac { 1 } { 2 } , and let (xi∗,yj∗) \left( x _ { i } ^ { * } , y _ { j } ^ { * } \right) be the center of Rij. Calculate the double Riemann sum of f.


A) 316\frac { 3 } { 16 }

B) 14\frac { 1 } { 4 }
C) 516\frac { 5 } { 16 }
D) 38\frac { 3 } { 8 }
E) 716\frac { 7 } { 16 }
F) 12\frac { 1 } { 2 }

G) 916\frac { 9 } { 16 }

H) 34\frac { 3 } { 4 }

I) F) and G)
J) D) and E)

Correct Answer

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

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

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Find the area of the part of the cone 4x2+4y2=z24 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=x2y = x ^ { 2 } .


A) 56\frac { 5 } { 6 }

B) 53\frac { 5 } { 3 }
C) 23\frac { 2 } { 3 }
D) 2
E) 56\frac { \sqrt { 5 } } { 6 }
F) 53\frac { \sqrt { 5 } } { 3 }

G) 25\frac { \sqrt { 2 } } { 5 }
H) 6

I) B) and D)
J) E) and G)

Correct Answer

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

Correct Answer

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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 by changing to cylindrical coordinates.

Correct Answer

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Rewrite ∬Rf(x,y)dA\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. 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.

Correct Answer

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


A) 1120\frac { 1 } { 120 }

B) 190\frac { 1 } { 90 }
C) 160\frac { 1 } { 60 }
D) 148\frac { 1 } { 48 }
E) 136\frac { 1 } { 36 }
F) 124\frac { 1 } { 24 }

G) 112\frac { 1 } { 12 }

H) 16\frac { 1 } { 6 }

I) B) and F)
J) C) and E)

Correct Answer

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Write ∬Rf(x,y)dA\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.

Correct Answer

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


A) π4\frac { \pi } { 4 }

B) π3\frac { \pi } { 3 }

C) π2\frac { \pi } { 2 }

D) 2π3\frac { 2 \pi } { 3 }

E) 3π4\frac { 3 \pi } { 4 }

F) π\pi

G) 4π3\frac { 4 \pi } { 3 }

H) 2 π\pi

I) C) and F)
J) C) and E)

Correct Answer

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Let f(x,y) =xyf ( x , y ) = x y , and let R={(x,y) ∣0≤x≤1,0≤y≤1}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=12x = \frac { 1 } { 2 } and y=12y = \frac { 1 } { 2 } , and let (xi∗,yj∗) \left( x _ { i } ^ { * } , y _ { j } ^ { * } \right) be the upper right corner of Rij. Calculate the double Riemann sum of f.


A) 316\frac { 3 } { 16 }

B) 14\frac { 1 } { 4 }
C) 516\frac { 5 } { 16 }
D) 38\frac { 3 } { 8 }
E) 716\frac { 7 } { 16 }
F) 12\frac { 1 } { 2 }

G) 916\frac { 9 } { 16 }

H) 34\frac { 3 } { 4 }

I) A) and H)
J) E) and H)

Correct Answer

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