Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. x = 1−y^4, x = 0; about x = 1.

Answer 1

using shells of thickness dx, and taking advantage of symmetry,

v = 2∫[0,1] 2πrh dx
where r=1-x and h=y=∜(1-x)
v = 2∫[0,1] 2π(1-x)∜(1-x) dx
= 4π∫[0,1] (1-x)^(5/4) dx = 16π/9

using discs (washers) of thickness dy, we have

v = 2∫[0,1] π(R^2-r^2) dy
where R=1 and r=1-x
v = 2∫[0,1] π(1-(1-(1-y^4))^2) dy
= 2∫[0,1] π(1-y^8) dy
= 16π/9