Question

Find the volume of the solid formed by revolving the region bounded by the graphs of y = x^3, x = 2, and y = 1 about the y-axis.

A. 93pi/5
B. 120pi/7
C. 47pi/5 <--- My choice
D. None of these

Am I right?

Please help, Thank you!

Answer 1

using shells of thickness dx,
v = ∫[1,2] 2πrh dx
where r=x and h=y-1 = x^3-1
v = ∫[1,2] 2πx(x^3-1) dx = 47π/5

using washers of thickness dy,
v = ∫[1,8] π(R^2-r^2) dy
where R=2 and r=x = ∛y
v = ∫[1,8] π(2^2-y^(2/3)) dy = 47π/5

good work