How do you use the disk method to find the volume of the solid formed by rotating the region bounded by #y = 2x# and #y = x^2# about the y-axis?

1 Answer
May 15, 2015

#y = 2x# and #y = x^2# intersect at #(0,0)# and at #(2,4)#

I can't get both curves on one graph, but I'll assume you can graph the line and the parabola.

Rotating around the #y# axis and using disks, means our independent variable will be #y#.

The representative slice is horizontal with left end (little radius: r) on the line #x=1/2y# and right end (big radius: R) on the parabola #x=sqrty#.
The thickness of the disk will be #dy#, and the limits of integration will be #0# to #4#.

#int_0^4 piR^2-pir^2 dy = piint_0^4 ((sqrty)^2-(y/2)^2) dy#

#pi int_0^4 (y-y^2/4) dy = pi[y^2/2-y^3/12]_0^4 =pi[8-16/3] =(8pi)/3#