Question

Let R be the region in the xy-plane between the graphs of y = ex and y = e-x from x = 0 to x = 2.
a) Find the volume of the solid generated when R is revolved about the x-axis.

Answer 1

1a) Washer because two functions are involved, both in terms of x and it's being revolved around a horizontal line (an equation in terms of x).
b) Two choices, either use shell or solve each equation for x and find new limits and proceed with washer in a different variables.

Answer 2

Start off \[\int\limits_{0}^{2}(1-e^(-x))^2\]

pi (Big radius)^2 - pi (small radius)^2 or
pi [ (Big radius)^2 - (small radius)^2 ]
[pi int_{0}^{2}e^{2x}-e^{-2x}dx\]
\[(e^x)^2=e^x*e^x=e^{x+x}=e^{2x}\]
\[(e^-x)^2=e^-x*e^-x=e^{-x-x}=e^{-2x}\
[\pi(\frac{e^{2(2)}}{2}+\frac{e^{-2(2)}}{2})-\pi(\frac{e^{2(0)}}{2}+\frac{e^{-2(0)}}{2})\]
Should come out to 26pi