In-class work for 5 July 2012

1. Evaluate the iterated integral.
    a.) $\displaystyle\int_0^1 \int_x^{2x} \int_0^t 2xyz dzdydx$
    b.) $\displaystyle\int_0^1 \int_0^z \int_0^y ze^{-y^2} dxdydz$

2. Evaluate the triple integral.
    a.) $\displaystyle\int\int\int_E yz \cos(x^5) dV$; $E = \{(x,y,z) \colon 0 \leq x \leq 1, 0 \leq y \leq x, x \leq z \leq 2x\}$
    b.) $\displaystyle\int\int\int_E xy dV$; $E$ is bounded by the parabolic cylinders $y=x^2$ and $x=y^2$ and the planes $z=0$ and $z=x+y$
    c.) $\displaystyle\int\int\int_T xyz dV$; $T$ is the solid tetrahedron with vertices $(0,0,0), (1,0,0), (0,1,0),$ and $(1,0,1)$

3. Use a triple integral to find the volume of the given solid.
    a.) The solid bounded by the cylinder $y=x^2$ and the planes $z=0, z=4$, and $y=9$.
    b.) The solid enclosed by the paraboloid $x=y^2+z^2$ and the plane $x=16$.

4. Sketch the solid whose volume is given by the iterated integral $\displaystyle\int_0^2 \int_0^{2-y} \int_0^{4-y^2} dx dz dy$.

5. Write five other iterated integrals that are equal to the iterated integral $\displaystyle\int_0^1 \int_0^{x^2} \int_0^y f(x,y,z) dz dy dx$.

6. Find the mass and center of mass of the solid $E$ bounded by the parabolic cylinder $z=1-y^2$ and the planes $x+z=1, x=0,$ and $z=0;$ with density function $\rho(x,y,z)=2$.