Set up a triple integral?

Question

Set up a triple integral that equals the volume of the intersection of the interior of two cylinders of radius a>0 whose axes intersect at a right angle. HINT: Take one of the cylinders to be: x^2+y^2=a^2, and the other one x^2+z^2=a^2.

Answer 1

Use symmetry,

V = 8 ∫{0, a} dx ∫{0, √(a² - x²) } dy∫{0, √(a² - x²)} dz

Answer: V = (16/3)a³

Answer 2

Using the hint provided gives us two cylinders which intersect about the origin. A key to making integration problem easier is to take advantage of symmetry when you can. In this case the bounded region is symmetric about the origin which means we need only consider one octant (and of course multiply the iterated integral by 8). From there it should be a lot easier to draw a picture to see what the limits of integration should be.

I hope this was at least somewhat helpful. If not I'll be glad to elaborate more.