Question

Let M be the region under the graph f(x) = 3/(e^x) from x = 0 to x = 5.

M is the base of a solid whose cross sections are semicircles whose diameter lies in the xy plane. The cross sections are perpendicular to the x-axis. Find the volume of this solid.
- I think M is equal to 3 - 3/(e^5)

Answer 1

It does not really matter what the area of M is, but you are correct.
each circle's diameter is a strip under the graph, so adding then all up,
v = ∫1/2 πr^2 dx
where r = y/2 = 3/2 e^-x
v = ∫[0,5]1/2 π(3/2 e^-x)^2 dx = 9π/16 (1 - e^-10)

Answer 2

area of cross section = (1/2) pi (D/2)^2 where D = 3/e^x
= pi D^2 / 8 =(9/8) pi / e^2x
so you want the integral from x = 0 to x = 5 of
(9 pi / 8) e^-2x dx
(9 pi / 8)(-1/2) e^-2x at 5 - at 0
e^-10 is zilch so just - at zero which is +
(9 pi / 8)(1/2)= 9 pi /16