How to find the volume of a solid bounded by x which is equal to an arbitrary constant

Question

(solved)

I stumbled on this question today and I'm extremely curious on how to solve it.

My main concern is that x = b (b>1). I don't understand how can I bound this if x= b and b is any value greater than 1. If I ignore that last restriction I can do it.

This is what I got ignoring the last restriction. Anyhow, how do you actually solve the problem for a and b? Please let me know.

Answer 1

First of all, you have the wrong region; you get a ring or stand-like object when you revolve around the Y axis, not the X axis. You should take a rectangle-ish region to the right of the one you selected, going from 1 out to b, and then swing that in a horizontal circle. Second, I don't know if you need to do this by one particular method, washers versus cylinders. For washers, you need to solve the function for x, and integrate in two pieces...easier to use cylinders centered on the y axis with radii from 1 to b. The extra factor of 2$\pi$ x makes the integral more doable.

Answer 2

Use Cylindrical method. The volume is found by integrating (2 pi x f(x)) from x=1 to x=b, where f(x) is the given function. Your answer depends on b of course.