# Finding Volume with Integration

student92

## Homework Statement

How do you determine what formula to use in a given volume question? I always confused about it. Like, when to apply shell, when should use washer, etc.

For example:
Find the volume of the solid generated by revolving the area between the curve y=(cos x)/x and the x-axis for π/6≤x≤π/2 about the y-axis.

## Homework Equations

Should I use this one
$V = \int\limits_{a}^{b}2\pi xf(x)dx$
Or this one
$V=π\int\limits_{a}^{b}y^2dx$

## The Attempt at a Solution

Using first equation, I will get
$V = \int\limits_{\pi/6}^{\pi/2} 2\pi x \frac{cosx}{x}dx = 2\pi \int\limits_{\pi/6}^{\pi/2}cosxdx = 2\pi [sinx]_{\pi/6}^{\pi/2} = 2 \pi (1 - 1/2)= \pi$

I don't know how to calculate the second one but someone write the answer as follow
$\pi \int\limits_{\pi/6}^{\pi/2}(\cos x/x)^2dx= \pi (-x) \frac{(-x Si(2x)+\cos^2(x))}{(x)}|_\frac{\pi}{6}^\frac{\pi}{2}=Si(\frac{\pi}{3})-Si(\pi)+\frac{9}{2\pi}$

Those answers are not equal. I have checked with wolframalpha.

eg. divide the volume into disks along the x-axis ... the disk at position x has thickness dx and radius R=f(x) ... so it's volume is dV: the volume of a disk is the area of the surface times the thickness. Thus: dV=πR2dx = πf2(x)dx and the total volume will be the volumes of all these disks added up thus:$$V=\pi \int_a^b f^2(x)dx$$