Exercise 20

Let $D$ be the region that lies inside the unit circle in the plane.

1. Set up and evaluate an iterated integral in polar coordinates whose value is the area of $D$.

2. Determine the exact average value of $f(x,y) = y$ over the upper half of $D$.

3. Find the exact center of mass of the lamina over the portion of $D$ that lies in the first quadrant and has its mass density distribution given by $\delta(x,y) = 1$. (Before making any calculations, where do you expect the center of mass to lie? Why?)

4. Find the exact volume of the solid that lies under the surface $z = 8-x^2-y^2$ and over the unit disk, $D$.