Question

The base of a solid in the region bounded by the graphs of y = e-x y = 0, and x = 0, and x = 1. Cross sections of the solid perpendicular to the x-axis are semicircles. What is the volume, in cubic units, of the solid?
a) π /16 e^2
b) π /16 (1-1/e^2)
c) π/4(1-1/e^2)
d) π/16(e^2-1)

Answer 1

Answer:

b) π /16 (1-1/e^2)

Step-by-step explanation:

For this case we have the following limits:

$y =e^{-x} , y=0, x=0, x=1$

And we have semicircles perpendicular cross sections.

The area of interest is the enclosed on the picture attached.

So we are assuming that the diameter for any cross section on the region of interest have a diameter of $D=e^{-x}$

And then we can find the volume of a semicircular cross section with the following formula:

$V= \frac{1}{2}\pi (\frac{e^{-x}}{2})^2 dx= \frac{1}{8} \pi e^{-2x}$

And for th volum we can integrate respect to x and the limits for x are from 0 to 1, so then the volume would be given by this:

$V= \pi \int_0^{1} \frac{1}{8} \pi e^{-2x} dx$

$V= -\frac{\pi}{16} e^{-2x} \Big|_0^1$

And evaluating the integral using the fundamental theorem of calculus we got:

$V = -\frac{\pi}{16} (e^{-2} -1)= \frac{\pi}{16}(e^{-2} -1)=\frac{\pi}{16} (1-\frac{1}{e^2})$

And then the best option would be:

b) π /16 (1-1/e^2)