How do you express an iterated integral in six different ways?

Answer 1

First, draw the region/solid being bounded by parameters say:

y^2 + z^2 = 9, x = -2, and x = 2

Now analyze what possible iterated integrals can be used to find this region.

the two "main" iterated integrals are:

the triple integral from [-2,2] [-3,3] [-sqrt(9-y^2),sqrt(9-y^2)] dz dy dx

and [-2,2] [-3,3] [-sqrt(9-z^2),sqrt(9-z^2)] dy dz dx

Now, instead of sketching every region to find the different possible integrals, using the rules of triple integration, they will essentially be any legal alteration of the order of the "main" integrals.

essentially, the first main integral can be rewritten as dx dz dy, and dz dx dy

the second can be written as dx dy dz and dy dx dz.