The shell has the radius `x` , the cricumference is `2pi*x` and the height is `6x - 2x^2 - x^2` , hence, the volume can be evaluated, using the method of cylindrical shells, such that:
`V = 2pi*int_(x_1)^(x_2) x*(6x - 3x^2) dx`
You need to evaluate the endpoints `x_1` and `x_2` , such that:
`x^2 = 6x - 2x^2 =>3x^2 - 6x = 0 => 3x(x - 2) = 0 => 3x = 0 =>...
The shell has the radius `x` , the cricumference is `2pi*x` and the height is `6x - 2x^2 - x^2` , hence, the volume can be evaluated, using the method of cylindrical shells, such that:
`V = 2pi*int_(x_1)^(x_2) x*(6x - 3x^2) dx`
You need to evaluate the endpoints `x_1` and `x_2` , such that:
`x^2 = 6x - 2x^2 =>3x^2 - 6x = 0 => 3x(x - 2) = 0 => 3x = 0 => x = 0 and x - 2 = 0 => x = 2`
`V = 2pi*int_0^2 x*(6x - 3x^2) dx`
`V = 2pi*(int_0^2 6x dx - int_0^2 3x^2 dx)`
Using the formula `int x^n dx = (x^(n+1))/(n+1)` yields:
`V = 2pi*(6x^2/2 - 3x^3/3)|_0^2`
`V = 2pi*(3x^2 - x^3)|_0^2`
`V = 2pi*(3*2^2 - 2^3)`
`V = 2pi*4`
`V = 8pi`
Hence, evaluating the volume, using the method of cylindrical shells, yields `V = 8pi.`
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