Thread: Need help on "Volume by Cylindrical Shells"

1. Need help on "Volume by Cylindrical Shells"

I have this question due, but I don't know how to solve it. Can someone please help me?

2. Re: Need help on "Volume by Cylindrical Shells"

What have you tried?

(a) Let's set it up. We are using volume by cylindrical shell. So, the volume is:

$\displaystyle V = \int_0^a 2\pi x y dx = \int_0^a 2\pi xf(x) dx$

Then the hint is $u =y= f(x), du = f'(x)dx = -xf(x)dx = -xudx$

So, $-xdx = \dfrac{du}{u}$ gives:

$\displaystyle V = \int_{f(0)}^{f(a)} -2\pi du = 2\pi \int_{f(a)}^1 du = 2\pi (1-f(a))$

(b) the diameter is 2a. So, if a approaches infinity, so too does 2a.

(c) As a approaches infinity, the volume approaches $2\pi$, as $f(a)$ approaches zero.