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Exam 15: Multiple Integrals

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Question 1

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Use polar coordinates to find the volume of the solid under the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ and above the disk $x ^ { 2 } + y ^ { 2 } \leq 4$ . Select the correct answer.

A) $64 \pi$
B) $5.3 \pi$
C) $32 \pi$
D) $16 \pi$
E) $8 \pi$

F) A) and E)
G) All of the above

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Question 2

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Find the area of the surface $S$ where $S$ is the part of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 16$ that lies inside the cylinder $x ^ { 2 } - 4 x + y ^ { 2 } = 0$ .

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Question 3

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Use cylindrical coordinates to evaluate the triple integral $\iiint _ { E } y d V$ where $E$ is the solid that lies between the cylinders $x ^ { 2 } + y ^ { 2 } = 3$ and $x ^ { 2 } + y ^ { 2 } = 7$ above the $x y$ -plane and below the plane $z = x + 4$ .

Evaluate the integral $\iiint _ { B } f ( x , y , z ) d V$ where $f ( x , y , z ) = x y ^ { 2 } + y z ^ { 2 }$ and $B = \{ ( x , y , z ) \mid 0 \leq x \leq 2 , - 1 \leq y \leq 1,0 \leq z \leq 3 \}$ with respect to $x , y$ , and $z$ , in that order Select the correct answer.

Use a double integral to find the area of the region $R$ where $R$ is bounded by the circle $r = 8 \sin \theta$ . Select the correct answer.

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 $y = x - 2$ . $\rho ( x , y ) = 3$ Select the correct answer.

Use cylindrical coordinates to evaluate $\iiint _ { B } \sqrt { x ^ { 2 } + y ^ { 2 } } d V$ where $E$ is the region that lies inside the cylinder $x ^ { 2 } + y ^ { 2 } = 25$ and between the planes $z = - 3$ and $z = 5$ . Round the answer to two decimal places.

Use cylindrical coordinates to evaluate the triple integral $\iiint _ { E } y d V$ where $E$ is the solid that lies between the cylinders $x ^ { 2 } + y ^ { 2 } = 3$ and $x ^ { 2 } + y ^ { 2 } = 7$ above the $x y$ -plane and below the plane $z = x + 4$ . Select the correct answer.

The sketch of the solid is given below. Given $a = 7$ , write the inequalities that describe it.

Describe the region whose area is given by the integral. $\int _ { 0 } ^ { x / 4 } \int _ { 0 } ^ { \cos 2 \theta } 2 r d r d \theta$

Calculate the double integral. Round your answer to two decimal places. $\iint _ { R } x y e ^ { y } d A , R = \{ ( x , y ) \mid 0 \leq x \leq 3,0 \leq y \leq 3 \}$

Use the transformation $x = \sqrt { 2 } u - \sqrt { \frac { 2 } { 3 } } v , y = \sqrt { 2 } u + \sqrt { \frac { 2 } { 3 } } v$ to evaluate the integral $\iint _ { R } \left( x ^ { 2 } - x y + y ^ { 2 } \right) d A$ , where $R$ is the region bounded by the ellipse $x ^ { 2 } - x y + y ^ { 2 } = 2$ .

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

Find the mass of the solid $S$ bounded by the paraboloid $z = 6 x ^ { 2 } + 6 y ^ { 2 }$ and the plane $z = 5$ if $S$ has constant density 3 .

Use polar coordinates to find the volume of the solid bounded by the paraboloid $z = 7 - 6 x ^ { 2 } - 6 y ^ { 2 }$ and the plane $z = 1$ . Select the correct answer.

Find the mass and the moments of inertia $I _ { x } , I _ { y }$ , and $I _ { 0 }$ and the radii of gyration $\bar { x }$ and $\bar { y }$ for the lamina occupying the region $R$ , where $R$ is the region bounded by the graphs of the equations $x = 2 \sqrt { y } , x = 0$ , and $y = 2$ , and having the mass density $\rho ( x , y ) = x y$ .

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 .

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 $x y = 2 , x y = 4 ; x = \frac { u } { v } , y = v$ .

Evaluate the iterated integral by converting to polar coordinates. Round the answer to two decimal places. $\int _ { - 3 } ^ { 3 } \int _ { 0 } ^ { \sqrt { 9 - y ^ { 2 } } } \left( x ^ { 2 } + y ^ { 2 } \right) ^ { 3 / 2 } d x d y .$

Find the mass of a solid hemisphere of radius 5 if the mass density at any point on the solid is directly proportional to its distance from the base of the solid.