Exam 15: Multiple Integrals

Using spherical coordinates, the triple integral $\iiint$ _E $e ^ { \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { \frac { 1 } { 2 } } } d V$ where E is the solid $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \leq 1$ is

The triple integral $\iiint$ _E $y ^ { 2 } d V$ where E is the region bounded by the surfaces $x ^ { 2 } + y = 1 , z ^ { 2 } + 2 y = 4 , y = 0$ is

Using spherical coordinates, the volume of the solid inside $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4 z$ and above $x ^ { 2 } + y ^ { 2 } = z ^ { 2 }$ is

The volume of the surface bounded by the graph of $z = e ^ { x + y }$ with $0 \leq x \leq 1,0 \leq y \leq 1$ is

Let and z = z. Then $x = r \cos \theta$ is

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

If the order of integration of $\int _ { 0 } ^ { 1 } \left[ \int _ { y ^ { 2 } } ^ { \sqrt [ 3 ] { y } } f ( x , y ) d x \right] d y$ is switched, the result is

The volume of the solid cut out of $z ^ { 2 } + r ^ { 2 } = 4$ by r = 1 is

The surface area of the surface $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ above z = 0 and inside $x ^ { 2 } + y ^ { 2 } = 2 x$ is

The triple integral $\iiint$ _E $\left( x ^ { 2 } + z ^ { 2 } \right) d V$ where E is the region bounded by the plane $12 x + 20 y + 15 z = 60$ and the coordinate planes is

The triple integral $\iiint$ _E $x y d V$ where E is the tetrahedron with vertices (0, 0, 0), (1, 1, 0), (1, 0, 0), and (1, 0, 1) is

The iterated integral $\int _ { 0 } ^ { \frac { \pi } { 4 } } \left[ \int _ { 0 } ^ { 1 } 2 x \cos ( y ) d x \right] d y$ is

Let $x = \frac { u + v } { 2 }$ and $y = \frac { u - v } { 2 }$ Then $\frac { \partial ( x , y ) } { \partial ( u , v ) }$ is

The iterated integral $\int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { \sqrt { 2 } } \int _ { r } ^ { \sqrt { 4 - r ^ { 2 } } } r d z d r d \theta$ 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 = 4, x = 1, and x = 4 is

The triple integral $\iiint$ _E $x d V$ where E is the region bounded by the plane $x + 2 y + 3 z = 6$ and the coordinate planes is

If $\rho ( x , y , z ) = x ^ { 2 } + y ^ { 2 } + z ^ { 2 }$ is the mass density of the solid bounded by $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ then its total mass is

Using cylindrical coordinates, the volume of the solid bounded by $x ^ { 2 } + y ^ { 2 } + z = 1$ and z = 0 is

The volume of the solid bounded by x = z, x = 8 - z, y = z, y = 8 and z = 0, z = 4 is

Using cylindrical coordinates, the volume of the solid bounded by $x ^ { 2 } + y ^ { 2 } = 4 y , x ^ { 2 } + y ^ { 2 } = 4 z$ and z = 0 is