Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.

Question

y = 3e^(-x), y = 3, x = 2;
about y = 6

Answer 1

Using discs (washers) of thickness dx,

v = ∫[0,2] π(R^2-r^2) dx
where R=6-y and r=3
v = ∫[0,2] π((6-3e^(-x))^2-3^2) dx
= 9π(5e^4+8e^2-1)/(2e^4)

using washers of thickness dy,
v = ∫[3/e^2,3] 2πrh dy
where r=6-y and h=2-x
v = ∫[3/e^2,3] 2π(6-y)(2+ln(y/3)) dy
= 9π(5e^4+8e^2-1)/(2e^4)