I have always seen the integral as the area under a curve. So for instance, if you integrated over the upper arc of a circle you would get ½[itex]\pi[/itex]R2. But then, I learnt to do integrals in spherical coordinates and something confuses me. If you do the integral from 0 to 2[itex]\pi[/itex], you don't get the area of a circle - you get the length of the circumference. Why is that? Certainly you are integrating over the arc of a circle? I can see, that you would need to integrate from 0 to R to actually get the formula for the area of a circle, but then you do a double integral, whereas in cartesian coordinates you would only integrate over y = ±√(R2-x2). Isn't there some sort of mismatch here? Edit: Yes, indeed when thinking of it, I am surprised that you do surface integrals as double integrals, when a single integral already yields and area?