Using the disk method

Question

Using the disk method to find the volume for y=sinx; bounded by the y=0, x=0 and x=pi; about the x axis. Also please demonstrate graphically!

Answer 1

V = π∫₀^πsin²xdx = π∫₀^π(1 - cos2x)/2dx = π(x/2 - sin2x/4)₀^π = π²/2

Answer 2

At x, draw a narrow vertical strip (of width dx) centered at x with base along the x-axis and height sinx. If we revolve the strip about the x-axis, we generate a solid disk with radius sinx and width dx.

Volume of disk = πsin²xdx

Volume of solid = π∫_{(from 0 to pi)} sin²xdx = (π/2)∫_{(from 0 to pi)} [(1 - cos(2x)]dx

= (π/2) [ x - (1/2)sin(2x) ] _{(from 0 to pi)} = π² / 2