In-class work for 29 June 2012
1.) Estimate the volume of the solid lying below the surface $z=xy$ and above the rectangle $R=\{(x,y) | 0 \leq x \leq 6, 0 \leq y \leq 4\}$. Use a Riemann sum with $m=3, n=2$ and take the sample point to be the upper right corner of each square.

2.) Evaluate the double integral by first identifying it as the volume of a solid:
    $\displaystyle\int\int_R 3 dA$, $R = \{(x,y) | -2 \leq x \leq 2, 1 \leq y \leq 6 \}$
    $\displaystyle\int\int_R (5-x) dA, R = \{(x,y) | 0 \leq x \leq 5, 0 \leq y \leq 3\}$
3.) Calculate the iterated integral
    a.) $\displaystyle\int_0^2 \int_0^{\frac{\pi}{2}} x \sin y dy dx$
    b.) $\displaystyle\int_{\frac{\pi}{6}}^{\frac{\pi}{2}} \int_{-1}^5 \cos y dx dy$

4.) Calculate the double integral
    a.) $\displaystyle\int\int_R (6x^2y^3 - 5y^4) dA$, $R = \{(x,y) | 0 \leq x \leq 3, 0 \leq y \leq 1 \}$
    b.) $\displaystyle\int\int_R \frac{1+x^2}{1+y^2} dA$, $R = \left[0, \dfrac{\pi}{6} \right] \times \left[0, \dfrac{\pi}{3} \right]$

5.) Find the volume of the solid that lies under the hyperbolic paraboloid $z=4+x^2-y^2$ and above the square $R = [-1,1] \times [0,2]$.

6.) Find the average value of $f(x,y)=x^2y$ over the rectangle given by vertices $(-1,0), (-1,5), (1,5),$ and $(1,0)$.