Question

find the volume of the solid of revolution generated when the region bounded by the curve y =x^2,the x-axis, and the lines x = 1 and x = 2 is revolved about the x-axis.

Answer 1

integral from x =1 to x = 2 of 2 pi x y dx
= 2 pi integral [ x^3 dx]
= 2 pi x^4/4 from 1 to 2
= pi /2 [ 16 - 1 ] = 15/2 pi

Answer 2

The volume of a disk is πr^2 h. So, with discs of thickness dx, the volume is
∫[1,2] πy^2 dx = ∫[1,2] πx^4 dx = π/5 x^5 [1,2] = 31π/5

Or, if you want to use shells, things get a bit more complicated. The volume consists of a solid cylinder of radius1 and height 1, plus a region bounded by the curve. The volume of a cylinder is 2πrh dy. That makes the whole thing
π + ∫[1,4] 2πy(2-√y) dy = 31π/5

Answer 3

Whoops, sorry, I spun it around the y axis. Use oobleck reply.