Exam 15: Multiple Integrals

By Fubini's Theorem, the double integral $\iint$ R $6 \left( x ^ { 2 } - y \right) d A$ with $R = \{ ( x , y ) : - 1 \leq x \leq 2,0 \leq y \leq 1 \}$ is

Let and $x = u + v$ Then $y = v + w$ is

The iterated integral $\int_{0}^{\frac{\pi}{2}} \int_{0}^{2 \cos ^{2} \theta } \int_{0}^{4-r^{2} } r \sin \theta d z d r d \theta$ is

The iterated integral $\int _ { 0 } ^ { 3 } \left[ \int _ { 0 } ^ { x } x ^ { 2 } e ^ { x y } d y \right] d x$ is

The center of mass of a lamina in the shape of a region in the xy-plane bounded by $r = \cos ( 2 \theta )$ for the petal on the right with area density $\rho ( r , \theta ) = r$ is

If R is the region bounded by $x ^ { 2 } + y ^ { 2 } = 4 , x ^ { 2 } + y ^ { 2 } = 9$ then $\iint$ _R $\left( x ^ { 2 } + y \right) d A$ in polar form is

If R is the region bounded by $x ^ { 2 } + y ^ { 2 } = 9$ then $\iint$ _{R $x ^ { 2 } \sqrt { 9 - x ^ { 2 } } d A$} is

The triple integral $\iiint$ _{E $y d V$} where E is the solid bounded by the plane $12 x + 20 y + 15 z = 60$ and the coordinate planes is

The iterated integral $\int _ { 0 } ^ { \pi } \int _ { 0 } ^ {\cos \theta } \int _ { 0 } ^{\sqrt { 1 - r ^ { 2 }} } r d z d r d \theta$ is

The surface area of the surface $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 9$ above z = 0 and inside $x ^ { 2 } + y ^ { 2 } = 3 x$ is

The center of mass of a lamina in the shape of a region in the xy-plane bounded by the y-axis, $3 x + 2 y = 18$ and y = 0 with area density $\rho ( x , y ) = x y$ is

The volume of the surface bounded by the graph of $z = \frac { 1 } { x } + \frac { 1 } { y }$ with $1 \leq x \leq 2,1 \leq y \leq 2$ is

The iterated integral $\int _ { 0 } ^ { 4 } \left[ \int _ { 0 } ^ { y } d x \right] d y$ is

Using a change of variables, the double integral $\iint$ _{R $\frac { x y } { 1 + x ^ { 2 } y ^ { 2 } } d A$} where R is the region bounded by xy = 1, xy = 5, x = 1, and x = 5 is

The iterated integral $\int _ { \frac { \pi } { 4 } } ^ { \frac { \pi } { 2 } } \int _ { 0 } ^ { \frac { \pi } { 2 } } \int _ { 0 } ^ { \cos \varphi } \rho ^ { 4 } \sin ^ { 3 } ( \varphi ) \cos \theta d \rho d \theta d \varphi$ is

The iterated integral $\int _ { 0 } ^ { 4 } \left[ \int _ { 0 } ^ { y } \sqrt { 9 + y ^ { 2 } } d x \right] d y$ is

The center of mass of a lamina in the shape of a region in the xy-plane bounded by $y = \sqrt { x }$ and y = x with area density $\rho ( x , y ) = x$ is

Using a change of variables, the double integral $\iint$ _{R $( x + y ) e ^ { x - y } d A$} where R is the region bounded by the quadrilateral with vertices (4, 0), (6, 2), (4, 4), and (2, 2) is

The area bounded by $y = x ^ { 3 }$ and $y = x ^ { 2 }$ is

The iterated integral $\int _ { 0 } ^ { \pi } \int _ { 0 } ^ { 4 } \int _ { 0 } ^ { 1 } r e ^ { z } d z d r d \theta$ is