Exam 12: Multiple Integrals

Suppose X and Y are random variables whose density function is given by $f ( x , y ) = \left\{ \begin{array} { c l } 0.2 e ^ { - ( 05 x + 0.4 y ) } & \text { if } x \geq 0 , y \geq 0 \\0 & \text { otherwise }\end{array} \right.$ is a joint density function.

Sketch the region E whose volume is given by the integral $\int _ { 0 } ^ { 2 x } \int _ { x / 6 } ^ { 5 x / 6 } \int _ { 1 / \sin \phi } ^ { 2 } \beta ^ { 2 } \sin \phi d \rho d \phi d \theta$ .

Use the change of variables $u = x y$ , $v = x y ^ { 2 }$ to evaluate $\iint _ { R } y ^ { 2 } d A$ , where R is the region bounded by the curves xy = 1, xy = 2, $x y ^ { 2 } = 1$ , and $x y ^ { 2 } = 2$ .

Use polar coordinates to evaluate $\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { \sqrt { 1 - x ^ { 2 } } } e ^ { - \left( x ^ { 2 } + y ^ { 2 } \right) } d y d x$ .

Find the z-coordinate of the centroid of the solid E bounded by the cone $z = \sqrt { x ^ { 2 } + y ^ { 2 } }$ and the plane $z = 2$ .

Let E be the part of the solid ellipsoid $x ^ { 2 } + y ^ { 2 } + 4 z ^ { 2 } \leq 9$ that lies in the first octant above the plane z = 1.(a) Express the triple integral $\iiint _ { Z } \frac { y } { z } d V$ as an iterated integral in rectangular coordinates.(b) Express the triple integral $\iiint _ { Z } \frac { y } { z } d V$ as an iterated integral in cylindrical coordinates.(c) Express the triple integral $\iiint _ { Z } \frac { y } { z } d V$ as an iterated integral in spherical coordinates.

Find the volume of the solid formed by the intersection of the cylinder $y = x ^ { 2 }$ and the two planes given by z = 0 and y + z = 4.

Find the volume of the solid bounded by the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ and the plane $z = 4$ .

Evaluate $\int _ { - 5 } ^ { 5 } \int _ { 0 } ^ { \sqrt { 25 - x ^ { 2 } } } \int _ { 0 } ^ { \sqrt { 25 - x ^ { 2 } - y ^ { 2 } } } \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { 1 / 3 } d z d y d x$ by changing to spherical coordinates.

Evaluate the iterated integral $\int _ { - 1 } ^ { 1 } \int _ { 0 } ^ { \sqrt { 1 - x ^ { 2 } } } \sin \left( \pi \left( x ^ { 2 } + y ^ { 2 } \right) \right) d y d x$ by converting to polar coordinates.

Find the area of that part of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ that lies above the plane z = 1.

Find the area of the part of the cylinder $y ^ { 2 } + z ^ { 2 } = 9$ that is above the rectangle R = [0, 2] $\times$ [3, 3].

Compute the area of that part of the graph of $3 z = 5 + 2 x ^ { 3 / 2 } + 4 y ^ { 3 / 2 }$ which lies above the rectangular region in the first quadrant of the xy-plane bounded by the lines x = 0, x = 3, y = 0, and y = 6.

Evaluate $\iint _ { D } \frac { 1 } { \sqrt { x ^ { 2 } + y ^ { 2 } + 1 } } d A$ , where D is the region that lies between the circles $x ^ { 2 } + y ^ { 2 } = 1$ and $x ^ { 2 } + y ^ { 2 } = 9$ .

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

Use a double integral to find the volume of the solid bounded by the planes $x + 4 y + 3 z = 12$ , $x = 0$ , $y = 0$ , and $z = 0$ .

Set up the triple integral for $f ( x , y , z ) = \pi x ^ { 3 }$ over the solid E with vertices (0, 0, 0), (0, 0, 1), (0, 1, 0), (0, 1, 1), (1, 1, 0), and (1, 1, 1).

Evaluate $\int _ { 1 } ^ { 2 } \int _ { 1 } ^ { x ^ { 2 } } \frac { x } { y } d y d x$ .

Evaluate the triple integral $\iiint _ { E } z d V$ , where E is the wedge in the first octant bounded by $y ^ { 2 } + z ^ { 2 } = 1$ , y = x, and the yz-plane.

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