Calculus: Early Transcendentals, Chapter 6, 6.2, Section 6.2, Problem 11

The volume of the solid obtained by rotating the region bounded by the curves y^2=x and y=x^2 about y = 1, can be evaluated using the washer method, such that:
V = int_a^b pi*(f^2(x) - g^2(x))dx
You need to find the endpoint of interval, hence, you need to solve for x the following equation, such that:
sqrt x = x^2 => x - x^4 = 0 => x(1 - x^3) = 0 => x = 0 and x = 1
You need to notice that x^2 < sqrt x on [0,1], such that:
V = int_0^1 pi*(((sqrt x) - 1)^2 - (x^2 - 1)^2)dx
V = pi*int_0^1 (x- 2sqrt x + 1)dx - pi*int_0^1 (x^4-2x^2 + 1)dx
V = (pi*x^2/2 - (4/3)pi*x^(3/2) + pi*x - pi*x^5/5 + 2pi*x^3/3 - pi*x)|_0^1
V = pi*1^2/2 - (4/3)pi*1^(3/2) - pi*1^5/5 + 2pi*1^3/3 - 0
V = pi/2 + (4pi)/3 - pi/5 - (2pi)/3
V = pi/2 + (2pi)/3 - pi/5
V = (15pi + 20pi - 6pi)/30
V = (29pi)/30
Hence, evaluating the volume of the solid obtained by rotating the region bounded by the curves y^2=x , y=x^2 about y = 1 , using the washer method, yields V = (29pi)/30.