Exam 15: Multiple Integrals

Let $I = \int _ { 0 } ^ { 2 } \left[ \int _ { 0 } ^ { \sqrt { 4 - x ^ { 2 } } } \sin \left( x ^ { 2 } + y ^ { 2 } \right) d y \right] d x$ Then I in polar form is

If R is the region bounded by $y = 2 x , y = \frac { x } { 2 }$ and $x = \frac { \pi } { 2 }$ then $\iint$ R $\sin x d A$ is

The rectangular coordinates of the given cylindrical coordinates $\left( 2 , \frac { \pi } { 6 } , - 5 \right)$ are

If R is the region bounded by y = x, y = 2, and xy = 1, then $\iint$ _R $\frac { y ^ { 2 } } { x ^ { 2 } } d A$ is

The rectangular coordinates of the given spherical coordinates $\left( 4 , \frac { \pi } { 3 } , \frac { 2 \pi } { 3 } \right)$ are

The center of mass of a lamina in the shape of a region in the xy-plane bounded by the y-axis, $8 y = x ^ { 2 }$ and y = 2 with area density $\rho ( x , y ) = y + 1$ is

The iterated integral $\int_{0}^{\frac{\pi}{4}} \int_{0}^{2 r} \int_{0}^{\cos \theta} r \sec ^{3}(\theta) d z d r d \theta$ is

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

Let $I = \int _ { 0 } ^ { 2 } \left[ \int _ { 0 } ^ { \sqrt { 4 - x ^ { 2 } } } d y \right] d x$ Then I in polar form is

By Fubini's Theorem, the double integral $\iint$ R $2 x \tan ( y ) d A$ with $R = \left\{ ( x , y ) : 0 \leq x \leq 1,0 \leq y \leq \frac { \pi } { 4 } \right\}$ is

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

The moment of inertia of a lamina in the shape of a region in the xy-plane bounded by y = sin x, the x-axis, from x = 0 to x =? with area density $\rho ( x , y ) = y$ about the y-axis is

The volume of the solid bounded by is

The iterated integral $\int _ { 0 } ^ { \pi } \left[ \int _ { 0 } ^ { \pi } 2 x \sin ( y ) d x \right] d y$ is

Using cylindrical coordinates, the volume of the solid in the first octant bounded by $x ^ { 2 } + y ^ { 2 } = 1$ and z = x is

The center of mass of a lamina in the shape of a region in the xy-plane bounded by the x-axis, $y = \sin x$ from x = 0 to x =?with area density $\rho ( x , y ) = y$ is

Using spherical coordinates, the triple integral $\iiint$ _E $x y z d V$ where E is the solid $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 16$ is

Let $x = \frac { u + v } { 4 }$ and $y = \frac { v - u } { 4 }$ Then $\frac { \partial ( x , y ) } { \partial ( u , v ) }$ is

The iterated integral $\int _ { 0 } ^ { \frac { \pi } { 2 } } \int _ { \frac { \pi } { 6 } } ^ { \frac { \pi } { 2 } } \int _ { 0 } ^ { 2 } \rho ^ { 3 } \sin ( \varphi ) \cos \theta d \rho d \varphi d \theta$ is

The volume of the solid bounded above by $z = r ^ { 2 }$ and below by $z = 2 r \sin \theta$ is