Calculus: Early Transcendentals, Chapter 6, 6.2, Section 6.2, Problem 12

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 curves y = e^(-x), y = 1, x = 2 , about y = 2, using washer method, yields V = pi*(3 + 9e^4)/(2e^4).