1. Compute .
 

The best way to proceed to by converting to spherical coordinates. The big clue is that the limits of integration for z are essentially a sphere with radius 3, which makes for very tough antidifferentiation in rectangular coordinates but converts very nicely to spherical. Looking at the top view we see that we really only have the portion to the right of the y axis, or in the (standard viewpoint) 3-d view the front top quarter of the sphere. Thus in spherical coordinates the integral is, which works out to .
 
 
 

2. Compute , where E is the region bounded by the cylinder y² + z² = 9 and the planes x=0, y=3x, and z=0 in the first quadrant.
 

The cylinder (solved for z, selecting the positive radical) forms the top boundary of our region and z=0 forms the bottom, so these can serve as limits for z. For x and y limits the top view is the best place to look. Essentially we're talking about the region bounded by x=0, y=3x, and y=3 (because that's the relevant part of where the cylinder crosses the plane z=0), so the integral should be , which works out to 27/8.
 
 

3. Write as a triple integral and compute the volume of the region bounded by y = 0, z = 0, and z = 9 - x² - y.
 

It's a weird shape -- try using Mathematica to graph it if you're curious. Basically it's a parabolic cylinder sloping downward along the y axis, cut off by two of the coordinate planes. The easiest setup for the integral is , which works out to 648/5.