`x = y^2, x = 1 - y^2` 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 x = 3)

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You need to evaluate the volume using the washer method, such that:

`V = pi*int_a^b(f^2(x) - g^2(x))dx`

You need first to determine the endpoints, hence you need to solve for y the following equation, such that:

`y^2 = 1 - y^2`

`2y^2 = 1 => y^2 = 1/2 => y_(1,2)...

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You need to evaluate the volume using the washer method, such that:

`V = pi*int_a^b(f^2(x) - g^2(x))dx`

You need first to determine the endpoints, hence you need to solve for y the following equation, such that:

`y^2 = 1 - y^2`

`2y^2 = 1 => y^2 = 1/2 => y_(1,2) = +-(sqrt2)/2`

`V = pi*int_(-(sqrt2)/2)^((sqrt2)/2) ((3 - y^2)^2 - (3 - 1 + y^2)^2)dy`

`V = pi*int_(-(sqrt2)/2)^((sqrt2)/2) (9 - 6y^2 + y^4 - 4 - 4y^2 - y^4)dy`

`V = pi*int_(-(sqrt2)/2)^((sqrt2)/2) (5 - 10y^2)dy`

`V = pi*(int_(-(sqrt2)/2)^((sqrt2)/2) 5dy - int_(-(sqrt2)/2)^((sqrt2)/2) 10y^2dy)`

`V = pi*(5y|_(-(sqrt2)/2)^((sqrt2)/2) - 10y^3/3|_(-(sqrt2)/2)^((sqrt2)/2))`

`V = pi*(5((sqrt2)/2+(sqrt2)/2) - 10/3(2sqrt2/8 + 2sqrt2/8))`

`V = pi*(5sqrt2 - 10*sqrt2/6)`

`V = (20*pi*sqrt2)/6`

`V = (10*pi*sqrt2)/3`

Hence, evaluating the volume of the solid obtained by rotating the region bounded by the given curves, about x = 3, yields `V = (10*pi*sqrt2)/3.`

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