I apologize if am getting annoying, but I'm afraid that I still do not completely understand what you are trying to use 1/n of the area of the semi-circle and stuff.

What you are using is basically polar coordinates integration given by

where the limits of the inner integral is from 0 to r and the limits of the outer integral is from 0 to pi for your area of the semicircle.

For your original integral

if you are using the substitution

and

,

then your integral would become

running from

to

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I suspect you are actually using

with the substitution

am I correct?

In that case your variable is

and (you claimed) the upper limit of the integral is

in your post...

however, from the limits of the original integral you have x = a for lower and x = b for upper limit. Now, the problem is that you are actually integrating with respect to y if my reasoning on your derivation of

is correct; if the limits of the integral of

is c to d, then we have

&

Do you think

&

?" alt="I=\int_a^{b}\sqrt{r^2-x^2}dx=r^2\int_0^{\frac{\pi}{n}}\cos^2(\theta)d\th eta

In your original post you said that

I suspect you are actually using

with the substitution

am I correct?

In that case your variable is

and (you claimed) the upper limit of the integral is

in your post...

however, from the limits of the original integral you have x = a for lower and x = b for upper limit. Now, the problem is that you are actually integrating with respect to y if my reasoning on your derivation of

is correct; if the limits of the integral of

is c to d, then we have

&

Do you think

&

?" />