Integral question?

Using Cylindrical Shells, the volume equals
∫(x = 0 to 1) 2π * x * 11e^(-x²) dx
(the radius of a shell is x - 0 = x and the height is 11e^(-x²) - 0 = 11e^(-x²).)

To evaluate this, let w = -x² and dw = -2x.

Then, ∫(x = 0 to 1) 2π * x * 11e^(-x²) dx
= ∫(w = 0 to -1) 11πe^w * -dw
= ∫(w = -1 to 0) 11πe^w dw
= 11πe^w {for w = -1 to 0}
= 11π(1 - e^(-1)).

I hope this helps!

int(11e^(-x^2)x)dx
(-1/2)int(11e^(-x^2)(-2x))dx
(-11/2)e^(-x^2) evaluated from 0 to 1
(-11/2)/e - (-11/2) = 11/2 - 11/(2e)