In-class work for 20 July 2012

1. Evaluate the surface integral
    a.) $\displaystyle\int\int_S y dS$; $S$ is the surface $z = \frac{2}{3}(x^{3/2}+y^{3/2})$ $0 \leq x \leq 1; 0 \leq y \leq 1$
    b.) $\displaystyle\int\int_S yz dS$; $S$ is the surface with parametric equations $x=u^, y=u \sin v, z = u \cos v; 0 \leq u \leq 1, 0 \leq v \leq \frac{\pi}{2}$
    c.) $\displaystyle\int\int_S \sqrt{1+x^2+y^2} dS$, $S$ is the helicoid with vector equation $\vec{r}(u,v) = < u \cos v, u \sin v, v>; 0 \leq u \leq 1, u \leq v \leq \pi$
    d.) $\displaystyle\int\int_S x^2z^2 dS$; $S$ is part of the cone $z^2=x^2+y^2$ that lies between the planes $z=1$ and $z=3$
    e.) $\displaystyle\int\int_S (z+x^2y) dS$; $S$ is the part of the cylinder $y^2+z^2=1$ that lies between the planes $x=0$ and $x=3$ in the first octant

2. Evaluate the surface integral $\displaystyle\int\int_S \vec{F} \cdot d\vec{S}$ for the given vector field $\vec{F}$ and oriented surface $\vec{S}$
    a.) $\vec{F}(x,y,z) = < xze^y, -xze^y, z >$; $S$ is the part of the plane $x+y+z=1$ in the first octant and has downward orientation
    b.) $\vec{F}(x,y,z) = < x, -z, y >$; $S$ is the part of the sphere $x^2+y^2+z^2=4$ in the first octant with orientation toward the origin
    c.) $\vec{F}(x,y,z) = < xy, 4x^2, yz >$; $S$ is the surface $z=xe^y$, $0 \leq x \leq 1, 0 \leq y \leq 1$, with upward orientation
    d.) $\vec{F}(x,y,z) = < x, 2y, 3z >$; $S$ is the cube with vertices $(\pm 1, \pm 1, \pm 1)$
    e.) $\vec{F}(x,y,z) = < x^2, y^2, z^2 >$; $S$ is the boundary of the solid half-cylinder $0 \leq z \leq \sqrt{1-y^2}, 0 \leq x \leq 2$

3. Find the mass of a thin funnel in the shape of a cone $z = \sqrt{x^2+y^2}; 1 \leq z \leq 4$, if its density function is $\rho(x,y,z) = 10 - z$.

4. A fluid has density $870 kg/m^3$ and flows ina velocity field $\vec{v} = < y, x, 0 >$ where $x,y,$ and $z$ are measured in meters and the components of $\vec{v}$ are in meters per second. Find the rate of flow outward through the hemisphere $x^2+y^2+z^2=9$, $z \geq 0$.