`y = e^(-x^2), y = 0, x = 0, x = 1` Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the...

The shell has the radius x, the cricumference is `2pi*x` and the height is `e^(-x^2)` , hence, the volume can be evaluated, using the method of cylindrical shells, such that:

`V = 2pi*int_0^1 x*e^(-x^2) dx`

You need to use substitution method to solve the integral, such that:

`-x^2 = u => -2xdx = du => xdx = -(du)/2`

`V = 2pi*int_(u_1)^(u^2) e^u*(-du)/2`

`V = -pi*e^u|_(u_1)^(u^2)`

`V = -pi*e^(-x^2)|_0^1`

`V = -pi*(e^(-1^2) - e^(-0^2))`

`V = -pi*(1/e - e) => V = pi*(e - 1/e)`

`V = ((e^2-1)*pi)/e`

Hence, evaluating the volume, using the method of cylindrical shells, yields `V = ((e^2-1)*pi)/e.`