Question

Find the volumes of the solids obtained by rotating the region bounded in the first quadrant by y=x^3 and x=1 , x-axis rotating about x-axis and y-axis.

Answer 1

The curves intersect at (0,0) and (1,1), so

about the x-axis
using discs of thickness dx
v = ∫[0,1] πr^2 dx
where r = y = x^3
v = ∫[0,1] πx^6 dx

using shells of thickness dy
v = ∫[0,1] 2πrh dy
where r=1 and h=1-x
v = ∫[0,1] 2πy(1 - ∛y) dy
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about the x-axis
using discs (washers) of thickness dy
v = ∫[0,1] π(R^2-r^2) dy
where R=1 and r=x=∛y
v = ∫[0,1] π(1-y^(2/3)) dy

using shells of thickness dx
v = ∫[0,1] 2πrh dx
where r=x and h=y
v = ∫[0,1] 2πx*x^3 dx