Question

how do i use the shell method to find the volumes of the solids generated by revolving the regions bounded by the curves and lines
a) y=1/x, y=0, x=1/2, x=2

So I think the integral is 2pi(radius)(height) dx from 0.5 to 2 but I'm not sure what the radius and height would be. the answer is 3pi but i just can't get it no matter how much i try.

Answer 1

the height is the radius.
so y = 1/x ---> r = 1/x

Vol = pi [ integral]r^2 dx from 1/2 to 2

(notice the pi(r^2)dx is your shell or cylinder, with r as the radius, and dx as the width or height of the cylinger)

= pi [integral] 1/x^2 dx from 1/2 to 2
= pi (-1/x) from 1/2 to 2
= pi( -1/2 - (-1/(1/2) )
= pi(-1/2 + 2)
= (p/2)pi