Calculating the area of a curve rotating around an axis

Calculate the area of the rotational surface that arise when the curve rotates around the y-axis.

My approach is something along the lines of:

Use an integral to calculate the area.

Since the curve is rotating around the y-axis, we need to (not sure about the english term), "extract" x from the formula (we need to get the height between the curve and the y-axis).

Since the curve is symmetrical around the y-axis we don't have to care the negative x; we simply multiply by 2 to get the total area.

So, we now have the following

which simplified should be

Something isn't right though, and I can't figure out what.