You need to evaluate the volume of the solid obtained by the rotation of the region bounded by the curves `y = x^3 , y = x, x =0` , about x 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, x = 0 is given. The other endpoint can be evaluated by solving the following equation:
`x^3 = x...
You need to evaluate the volume of the solid obtained by the rotation of the region bounded by the curves `y = x^3 , y = x, x =0` , about x 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, x = 0 is given. The other endpoint can be evaluated by solving the following equation:
`x^3 = x => x^3 - x = 0 => x(x^2 - 1) = 0 => x = 0, x = 1, x = -1`
You may evaluate the volume
`V = pi*int_(-1)^0 (x^6 - x^2)dx + pi*int_0^1 (x^2 - x^6)dx`
`V = pi*int_(-1)^0 (x^6)dx - pi*int_(-1)^0 x^2 dx + pi*int_0^1 x^2 dx - pi*int_0^1 x^6 dx`
`V = pi*((x^7)/7 - x^3/3)|_(-1)^0 + pi*(x^3/3 - x^7/7)|_0^1`
`V = pi*((0^7)/7 - 0^3/3 - 1/7 + 1/3 ) + pi*(1^3/3 - 1^7/7 - 0)`
`V = (4pi)/21 + (4pi)/21`
`V = (8pi)/21`
Hence, evaluating the volume of the solid obtained by the rotation of the region bounded by the curves `y = x^3 , y = x, x =0` , about x axis, yields `V = (8pi)/21.`
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