`y^2 = x, x = 2y` 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 the y-axis)

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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...

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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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