You need to evaluate the volume of the solid obtained by the rotation of the region bounded by the curves `y = x^3 , y = 0, x = 1` , about x = 2, using washer method, such that:
`V = int_a^b (f^2(x) - g^2(x))dx, f(x)>g(x)`
You need to find one endpoint, hence you need to solve the following equation:
`x^3 = 0=> x = 0`
You may evaluate the volume
`V = pi*int_0^1 (2 - root(3) y)^2dy`
`V = pi*int_0^1 (4 - 4 root(3) y + root(3) (y^2))dy`
`V = pi*(int_0^1 4dy - 4*int_0^1 y^(1/3)dy + int_0^1 y^(2/3) dy)`
`V = pi*(4y - 4*(3/4)*y^(4/3) + (3/5) y^(5/3))|_0^1`
`V = pi*(4y - 3*y^(4/3) + (3/5) y^(5/3))|_0^1`
`V = pi*(4 - 3 + (3/5) )`
`V = pi*(1 + 3/5)`
`V = (8pi)/5`
Hence, evaluating the volume of the solid obtained by the rotation of the region bounded by the curves `y = x^3 , y = 0, x = 1` , about x = 2, yields `V = (8pi)/5` .
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