Exam 12: Multiple Integrals

Evaluate the triple integral $\iiint _ { E } r d V$ in cylindrical coordinates, where $E = \{ ( r , \theta , z ) \mid 0 \leq r \leq 1,0 \leq \theta \leq 2 \pi , 0 \leq z \leq 1 \}$ .

The centroid (x, y, z) of a region E in $\mathbb { R } ^ { 3 }$ is the center of mass with the density function $\sigma ( x , y , z ) = 1$ . Find the centroid of the tetrahedron bounded by the coordinate planes and the plane x + y + z = 1.

Show that the surface areas for the functions $f ( x , y ) = 2 x ^ { 2 } + 2 y ^ { 2 }$ and $g ( x , y ) = 4 x y$ over the disk $D = \left\{ ( x , y ) \mid x ^ { 2 } + y ^ { 2 } \leq 1 \right\}$ are equal.

Evaluate the iterated integral $\int _ { 0 } ^ { 2 } \int _ { 0 } ^ { \sqrt { 2 y - y ^ { 2 } } } \frac { 1 } { \sqrt { x ^ { 2 } + y ^ { 2 } } } d x d y$ by converting to polar coordinates.

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

Evaluate $\iint _ { R } \sqrt { 4 - x ^ { 2 } } d A$ where $R = [ - 2,2 ] \times [ 0,3 ]$ by first identifying it as the volume of a solid.

Compute $\int _ { 0 } ^ { 2 } \int _ { - 1 } ^ { 1 } \frac { y e ^ { y ^ { 2 } } } { 1 + x ^ { 2 } } d x d y$ ..

Find the area of the surface with vector equation r $( u , v ) = \left\{ u \cos v , u \sin v , u ^ { 2 } \right\}$ , $0 \leq u \leq 2$ , $0 \leq v \leq 2 \pi$ .

Describe the image R of the set $S = \{ ( u , v ) \mid 0 \leq u \leq 1,0 \leq v \leq 1 \}$ under the transformation $x = 3 u + v$ , $y = u + 2 v$ , and then compute $\iint _ { R } \left( x y + y ^ { 2 } \right) d A$ .

Evaluate the double integral of $y e ^ { y ^ { 4 } }$ over the region bounded by $y = \sqrt { x }$ , y = 2, and x = 0.