Can the lower limit of the radius be variable, or must it always start at the or

Question

Can the lower limit of the radius be variable, or must it always start at the origin? Please explain in detail why or why not (this is the cruitial part of my question). You can reference a known theorm or proof to help explain your answer.

I get the correct answer to a problem when I have a variable lower limit for the radius, but my teacher said the radius always has to have a lower limit starting at the origin. My radius was variable in this problem, but still gave the correct result.

rdr dz dtheta Use cylindrical coordinates to set up the triple integral to find the volume of the solid inside x2 +y2 + z2 = 16 and outside z = .

Explanation / Answer

Yes, of course the radius can start at a length larger than zero. If you calculate the volume of a tall washer, you could do it in two steps, first calulating the volume of the large solid cylinder and then subtracting the volume of the hole, but that would be silly. In the two-step integral, both the solid cylinder and the hole start with radius = 0. If you write those two simple integrals side by side, you will see that the parts evaluated at r = 0 cancel (-0- -0), and we are left with the integral evaluated at the two r lengths that you suggest. If common sense isn't enough, I think the fact that there is no theory showing that you must start at zero radius is proof that you can start at a radius greater than zero.

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