semicircle problem

HallsofIvy

You haven't stated a problem here! What do you want to integrate from 0 to R? What function are you integrating?

If you just want the area, look at the problem you gave:
(And arildno did not say he had never seen that before, he was trying to get you to think about why you think that was true.)

Suppose your semi-circles radius is R. Then dA= r dr dθ. In a sense, dr here, as well as r, is r: along each radius we go from r= 0 to R so the "change in r", dr, is R.

Okay, then r dr dθ becomes dA= R² dθ, just like your dX dY became 4dy, the area of the semi-circle is
[tex]\int_{\theta=0}^{\pi} R^2 d\theta= \pi R^2[/tex], exactly the right answer.

(θ does not go from 0 to 180! As I am sure you learned in when you were learning the derivatives of sine and cosine, in calculus, all angles are in radians.)

(Oh, and by the way, why was this posted under "differential equations". Was this part of a differential equations problem?)