`y = e^(-x), y = 1, x = 2` Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk...

`y = e^(-x), y = 1, x = 2` Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. (about y = 2)

Expert Answers
sciencesolve eNotes educator| Certified Educator

You need to find the volume of the solid obtained by rotating the region enclosed by the curves `y = e^(-x), y = 1, x = 2` , about y = 2, using washer method:

`V = pi*int_a^b (f^2(x) - g^2(x))dx, f(x)>g(x)`

You need to find the endpoints by solving the equation:

`e^(-x)= 1 => 1/(e^x) = 1 => e^x = 1 => e^x = e^0 => x = 0`

`V = pi*int_0^2((e^(-x) - 2)^2 - (1 - 2)^2)dx`

`V = pi*int_0^2 (e^(-2x) - 4e^(-2x) + 4 - 1)dx`

`V = pi*int_0^2 (e^(-2x) - 4e^(-2x) + 3)dx`

`V = pi*(-(e^(-2x))/2 + 2e^(-2x) + 3x)|_0^2`

`V = pi*(-(e^(-4))/2 + 2e^(-4) + 6 + 1/2 - 2 + 0)`

`V = pi*(-1/(2e^4) + 2/(e^4)+ 4 + 1/2)`

`V = pi*(-1 + 4 + 9e^4)/(2e^4)`

`V = pi*(3 + 9e^4)/(2e^4)`

Hence, evaluating the volume of the solid obtained by rotating the region enclosed by the curvesĀ `y = e^(-x), y = 1, x = 2` , about y = 2, using washer method, yields `V = pi*(3 + 9e^4)/(2e^4).`