Calculus: Early Transcendentals, Chapter 6, 6.2, Section 6.2, Problem 9

The volume of the solid obtained by rotating the region bounded by the curves y^2=x and x=2y about y axis, 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 y the following equation, such that:
y^2 = 2y => y^2 - 2y = 0 => y(y-2) = 0 => y = 0 and y = 2
You need to notice that y^2 < 2y on [0,2], such that:
V = int_0^2 pi*(((2y)^2 - 0^2) - (y^4 - 0^2))dy
V = pi*int_0^2 4y^2dy - pi*int_0^2 (y^4)dy
V = (4pi*y^3/3 - pi*y^5/5)|_0^2
V = (4pi*2^3/3 - pi*2^5/5 - 4pi*0^3/3 + pi*0^5/5)
V = 32pi/3 - 32pi/5
V = 5*32pi/15 - 3*32pi/15
V = (64pi)/15
Hence, evaluating the volume of the solid obtained by rotating the region bounded by the curves y^2=x and x=2y about y axis, using the washer method, yields V = (64pi)/15.

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