calculus
posted by maggie
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.

Steve
using shells of thickness dx, and taking advantage of symmetry,
v = 2∫[0,1] 2πrh dx
where r=1x and h=y=∜(1x)
v = 2∫[0,1] 2π(1x)∜(1x) dx
= 4π∫[0,1] (1x)^(5/4) dx = 16π/9
using discs (washers) of thickness dy, we have
v = 2∫[0,1] π(R^2r^2) dy
where R=1 and r=1x
v = 2∫[0,1] π(1(1(1y^4))^2) dy
= 2∫[0,1] π(1y^8) dy
= 16π/9
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