Calculus: Early Transcendentals, Chapter 6, 6.2, Section 6.2, Problem 7

You need to evaluate the volume of the solid obtained by the rotation of the region bounded by the curves y = x^3 , y = x, x =0 , about x axis, using washer method, such that:
V = int_a^b (f^2(x) - g^2(x))dx, f(x)>g(x)
You need to find the next endpoint, since one of them, x = 0 is given. The other endpoint can be evaluated by solving the following equation:
x^3 = x => x^3 - x = 0 => x(x^2 - 1) = 0 => x = 0, x = 1, x = -1
You may evaluate the volume
V = pi*int_(-1)^0 (x^6 - x^2)dx + pi*int_0^1 (x^2 - x^6)dx
V = pi*int_(-1)^0 (x^6)dx - pi*int_(-1)^0 x^2 dx + pi*int_0^1 x^2 dx - pi*int_0^1 x^6 dx
V = pi*((x^7)/7 - x^3/3)|_(-1)^0 + pi*(x^3/3 - x^7/7)|_0^1
V = pi*((0^7)/7 - 0^3/3 - 1/7 + 1/3 ) + pi*(1^3/3 - 1^7/7 - 0)
V = (4pi)/21 + (4pi)/21
V = (8pi)/21
Hence, evaluating the volume of the solid obtained by the rotation of the region bounded by the curves y = x^3 , y = x, x =0 , about x axis, yields V = (8pi)/21.