Area of a Surface of Revolution

Question

The given curve is rotated about the y-axis. Set up, but do not evaluate, an integral for the area of the resulting surface by integrating with respect to x .

x = e^9y,    0 ≤ y ≤ 2

∫ (from 1 to e¹⁸) ( ? ) dx

Answer 1

That seems like a very strange way to evaluate the integral, but I suppose first, we'd find the change in the volume WRT (with respect to) the change in x.

So, the derivative of the volume WRT y is just πx². The derivative of x WRT y is 9e^(9y) = 9x. So, that's just πx/9.

So, ∫ [1, e¹⁸] πx/9 dx