In-class work for 19 July 2012
1. Determine whether the points $P$ and $Q$ lie on the given surface.
    a.) $\vec{r}(u,v) = < 2u+3v, 1+5u-v,2+u+v >$; $P=(7,10,4)$, $Q=(5,22,5)$
    b.) $\vec{r}(u,v) = < u+v, u^2-v, u +v^2 >$; $P=(3,-1,5), Q=(-1,3,4)$

2. Identify the surface with the given vector equation
    a.) $\vec{r}(u,v) = < u, u \cos v, u \sin v >$
    b.) $\vec{r}(y,z) = < 5y^2+2z^2-10, y, z >$


3. Find a parametric representation of
    a.) the plane passing thru $(1,2,-3)$ and containing the vectors $<1,1,-1>$ and $<1,-1,1>$.
    b.) the part of the plane $z=x+3$ that lies inside the cylinder $x^2+y^2+1$.

4. Find an equation of the tangent plane to the given parametric surface at the given point.
    a.) $x=u^2, y=v^2, z=uv; u=1, v=1$
    b.) $\vec{r}(u,v) = < uv, u \sin v, v \cos u >$; $u=0, v=\pi$

5. Find the area of the surface.
    a.) The part of the plane $2x+5y+z=10$ that lies inside the cylinder $x^2+y^2=9$.
    b.) The part of the surface $z=1+3x+2y^2$ that lies above the triangle with vertices $(0,0), (0,1),$ and $(2,1)$.
    c.) The "helicoid" $\vec{r}(u,v) = < u \cos v, u \sin v, v >$; $0 \leq u \leq 1, 0 \leq v \leq \pi$

6. Evaluate the surface integral.
    a.) $\displaystyle\int\int_S xy dS$; $S$ is the triangular region with vertices $(1,0,0), (0,2,0), (0,0,2)$
    b.) $\displaystyle\int\int_S y dS$; $S$ is the surface $z = \frac{2}{3}(x^{\frac{3}{2}}+y^{\frac{3}{2}}); 0 \leq x \leq 1, 0 \leq y \leq 1$
    c.) $\displaystyle\int\int_S yz dS$; $S$ is the part of the plane $x+y+z=1$ that lies in the first octant