Question

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

A. Find the area of M.

B. Find the value of c so that the line x=c divides the region M into two pieces with equal area.

C. 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.

Answer 1

A
a = ∫[0,5] 3/e^x dx = 3 - 3/e^5

B now you want c such that

∫[0,c] 3/e^x dx = ∫[c,5] 3/e^x dx
3 - 3/e^c = 3/e^c - 3/e^5
6/e^c = 3 + 3/e^5
e^c = 2/(1+e^-5)
c = ln 2/(1+e^-5)

C
each semicircle has diameter equal to y. Adding up all those thin slices of thickness dx, we have

v = ∫[0,5] π/4 9/e^(-2x) dx = 9π/8 (1-e^-10)