Question

Find the volume that is generated by rotating the area bounded by y=x^2 and y=2-x^2 about the x-axis using the shell method.

Answer 1

so, the shells are of thickness dy, and since the region is symmetric about the y-axis, the volume is
2∫[0,1] 2πrh dy
where r=y and h changes from √y to √(2-y) at y=1
v = 2∫[0,1] 2πy√y dy + 2∫[1,2] 2πy√(2-y) dy = 16π/3

check, using discs of thickness dx,
v = 2∫[0,1] π(R^2-r^2) dx
where R = 2-x^2 and r = x^2
v = 2∫[0,1] π((2-x^2)^2 - (x^2)^2) dx = 16π/3