Exam 15: Multiple Integrals

Using spherical coordinates, the volume of the solid bounded by the cone $\varphi = \frac { 2 \pi } { 3 }$ and the sphere $\rho = 6$ is

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

The iterated integral $\int _ { 0 } ^ { \frac { \pi } { 2 } } \int _ { 0 } ^ { 2 } \int _ { 0 } ^ { \sqrt { 16 - r ^ { 2 } } } r ^ { 2 } d z d r d \theta$ is

If $\sigma ( x , y , z ) = k$ is the volume density of the solid bounded by $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ then its moment of inertia about the z-axis is

Let $I = \int _ { - 3 } ^ { 3 } \left[ \int _ { 0 } ^ { \sqrt { 9 - x ^ { 2 } } } \left( x ^ { 2 } + y ^ { 2 } \right) d y \right] d x$ Then I in polar form is

The area bounded by $x ^ { 2 } + y ^ { 2 } = 16$ and $y ^ { 2 } = 6 x$ is

The iterated integral $\int _ { 0 } ^ { 1 } \left[ \int _ { 0 } ^ { 1 } e ^ { x } d x \right] d y$ is

The volume of the solid above the polar plane bounded by z = 2r and $r = 1 - \cos \theta$ is

The center of mass of a lamina in the shape of a region in the xy-plane bounded by the $x ^ { 2 } + y ^ { 2 } = 1 , x \geq 0 , y \geq 0$ with area density $\rho ( x , y ) = x + y$ is

The surface area of the surface cut from $2 x + y - z + 5 = 0$ by x = 0, x = 1, y = 0, and y = 4 is

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

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

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

The center of mass of a lamina in the shape of a region in the xy-plane bounded by $x ^ { 2 } + y ^ { 2 } = 4 , x \geq 0 , y \geq 0$ and x + y = 2 with area density $\rho ( x , y ) = x y$ is

The iterated integral $\int _ { 1 } ^ { 4 } \left[ \int _ { x ^ { 2 } } ^ { x } \sqrt { \frac { y } { x } } d y \right] d x$ is

The iterated integral $\int _ { 1 } ^ { e } \left[ \int _ { 0 } ^ { 2 } \frac { x } { y } d x \right] d y$ is

The volume of the solid in the first octant bounded by and all the coordinate planes is

If R is the region bounded by $x ^ { 2 } + y ^ { 2 } = 1 , x ^ { 2 } + y ^ { 2 } = 9$ then $\iint$ _R $5 \sqrt { x ^ { 2 } + y ^ { 2 } } d A$ in polar form is

The surface area of the surface $z = x ^ { 2 } + y ^ { 2 }$ below z = 4 is

The volume cut from the solid bounded $x ^ { 2 } + 4 y ^ { 2 } = 4$ by z = 0 and z = x + 2 is