Exam 13: Double and Triple Integrals

Find the mass of the solid whose density is equal to twice the distance from the origin, which is below the plane $z = 4$ and above the cone $z ^ { 2 } = x ^ { 2 } + y ^ { 2 }$ and the xy plane.

Find the volume between the graph of the given function and the specified rectangle. $f ( x , y ) = x ^ { 2 } y$ and $R = \{ ( x , y ) : - 1 \leq x \leq 1 \text { and } - 1 \leq \mathrm { y } \leq 1 \}$

Evaluate the sum $\sum _ { i = 1 } ^ { 3 } \sum _ { j = 1 } ^ { 2 } i ^ { 2 } j$

Write $y = 2$ in spherical coordinates.

Let $\rho ( r , \theta ) = \frac { \theta } { \pi ^ { 2 } }$ be a joint probability distribution function on the unit disk. What is the probability of an event occurring in the region bounded by the spiral $r = \frac { \theta } { 4 } , 0 \leq \theta \leq \pi$ and the x-axis?

Evaluate $\int _ { 0 } ^ { 2 } \int _ { 0 } ^ { 1 + x ^ { 2 } } \int _ { 1 - x + y } ^ { 1 + y } 2 z x d z d y d x$

Give the rectangular coordinates for the point with the cylindrical coordinates $\left( \sqrt { 2 } , \frac { \pi } { 4 } , 1 \right)$

Find the volume of the solid bounded by the graph of $z = \sqrt { 4 - x ^ { 2 } - y ^ { 2 } }$ and the xy plane.

Evaluate the integral $\int _ { 0 } ^ { \frac { \pi } { 2 } } \int _ { - \sin ( x ) } ^ { \sin ( x ) } \cos ( \cos ( x ) ) d y d x$

Find the mass of the solid in the first octant bounded by the coordinate planes and the plane $x + y + z = 1$ , where the density is $\rho ( x , y , z ) = 12 x y$

Let T be the triangle with vertices (0,0), (2,4), and (2,0). Let the density at each point of T be equal to the point's distance from the x-axis. Find the mass of T.

Use cylindrical coordinates to find $\iiint _ { R } d V$ , where $R = \left\{ ( r , \theta , z ) : 0 \leq \theta \leq \pi , 0 \leq \mathrm { r } \leq 4 \sin ( \theta ) , \text { and } 0 \leq \mathrm { z } \leq \sqrt { 16 - r ^ { 2 } } \right\}$

Evaluate the sum $\sum _ { i = 1 } ^ { 3 } \sum _ { j = 1 } ^ { 3 } \sum _ { k = 1 } ^ { 2 } ( k - j )$

Evaluate the sum $\sum _ { i = 1 } ^ { 2 } \sum _ { j = 1 } ^ { 3 } \sum _ { k = 1 } ^ { 2 } \frac { i j } { k }$

Change $x ^ { 2 } + y ^ { 2 } = z$ into cylindrical coordinates.

Find the signed volume between the graph of the given function and the specified rectangle. $f ( x , y ) = x ^ { 2 } y ^ { 3 }$ and $R = \{ ( x , y ) : 1 \leq x \leq 3 \text { and } - 1 \leq \mathrm { y } \leq 2 \}$

Use cylindrical coordinates to find $\iiint _ { R } z d V$ , where $R = \left\{ ( x , y , z ) : x ^ { 2 } + y ^ { 2 } \leq 4 , \text { and } 0 \leq \mathrm { z } \leq x ^ { 2 } + y ^ { 2 } \right\}$

Write $y = 2$ in cylindrical coordinates.

The functions $x = x ( u , v )$ and $y = y ( u , v )$ are given to determine transformations from the xy-coordinate system to a uv-coordinate system. Find the Jacobian of the transformation. x=2u-v y=2u+v

Evaluate the following $\iiint _ { R } x y z ^ { 2 } d V$ , where $R = \{ ( x , y , z ) : 0 \leq x \leq 1,0 \leq y \leq 3 , \text { and } - 1 \leq z \leq 1 \}$