`x = 2sqrt(y), x = 0, y = 9` 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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You need to evaluate the volume of the solid obtained by the rotation of the region bounded by the curves `x = 2sqrt y, x =0` , the line y = 9, about y axis, using washer method, such that:

`V = int_a^b (f^2(x) - g^2(x))dx, f(x)>g(x)`

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You need to evaluate the volume of the solid obtained by the rotation of the region bounded by the curves `x = 2sqrt y, x =0` , the line y = 9, about y 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, y = 9 is given. The other endpoint can be evaluated by solving the following equation:

`2sqrt y = 0 => 4y = 0 => y = 0`

You may evaluate the volume

`V = pi*int_0^9 ((2sqrt y)^2)dy`

`V = pi*int_0^9 (4y)dy`

`V = 4pi*y^2/2|_0^9`

`V = 2pi*y^2|_0^9`

`V = 2pi*(9^2 - 0^2)`

`V = 2pi*(81)`

`V = 162pi`

Hence, evaluating the volume of the solid obtained by the rotation of the region bounded by the curves `x = 2sqrt y, x =0` , the line y = 9, about y axis, yields `V = 162pi` .

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