3 Calculus III

In this class,

• Questions must be answered in order asked.
• Solutions must be neat.
• Solutions must be exact and fully simplified.
• You must use the given symbols.
• You must show all reasoning.
• Copying is never allowed, even when working together.

1. Find $I_x$ for the area between the curves
   
   $$y = x \qquad y = 4x -x^2$$
   
   
   Exact answers only, please. Since the integrand $y^2$ does not depend on $x$, it would seem logical to integrate $x$ first. Comment on that. [1, Centroids and Moments of Inertia]
2. Find the volume of the region bounded by
   
   $$z=0 \qquad x^2 + y^2 = 4x \qquad x^2 + y^2 = 4 z$$
   
   
   Use cylindrical coordinates $r$, $\theta$ (or $\phi$ if you want), and $z$ around the given $z$-axis.

   List the limits if (a) you do $z$ first, (b) you do r first, and (c) you do $\theta$ first. To do the latter two cases, make a picture of the cross-section of the region for a fixed value of $z$ like $z=\frac14$ and show the $r$ and $\theta$ integration lines.

   What variable is obviously the one to integrate first? For the second integration, discuss each possibility and explain which is the best choice. Use pictures to make your points.[1, Triple Integrals]

3. Try to do the previous question using Cartesian coordinates $x$, $y$ and $z$ instead of cylindrical ones. Work the volume out at least as far as a single-variable integral, and find the relevant parts in the Math handbook to find its anti-derivative. Use pictures to make your points.