Base of a solid

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?

 

π /16 e^2

 

π /16 (1-1/e^2)

 

π/4(1-1/e^2)

 

π/16(e^2-1)

Answer 1

First, draw the region.  

 

A typical cross section has diameter e^-x

 

Volume of semicircular cross section = (1/2) π(e^-x/2)²dx

 

                                                   = π(e^-2x/8)dx

 

Volume of solid = π∫_{(from 0 to 1)}(e^-2x/8)dx

 

                       = -π/16(e^-2x) _{(from 0 to 1)} 

 

                       = -π/16(e^-2 - 1)

 

                       = π/16(1 - 1/e²)

 

                       ≈ 0.17