Exercise 21

Consider the solid that is given by the following description: the base is the given region $D$, while the top is given by the surface $z = p(x,y)$. In each setting below, set up, but do not evaluate, an iterated integral whose value is the exact volume of the solid. Include a labeled sketch of $D$ in each case.

1. $D$ is the interior of the quarter circle of radius 2, centered at the origin, that lies in the second quadrant of the plane; $p(x,y) = 16-x^2-y^2$.

2. $D$ is the finite region between the line $y = x + 1$ and the parabola $y = x^2$; $p(x,y) = 10-x-2y$.

3. $D$ is the triangular region with vertices $(1,1)$, $(2,2)$, and $(2,3)$; $p(x,y) = e^{-xy}$.

4. $D$ is the region bounded by the $y$-axis, $y = 4$ and $x = \sqrt{y}$; $p(x,y) = \sqrt{1 + x^2 + y^2}$.