Originally Posted by

**Mathstud28** Hello everyone, here is my question. Now my brain may be muddled after thirty two hours without sleep, but

Consider for a second

$\displaystyle I=\int_a^{b}\sqrt{r^2-x^2}$

$\displaystyle I$ is describing the area under a portion of a semi-circle of radius $\displaystyle r$.

For sake of numerics but not straying completely away from calculus, instead of doing a nasty trig sub, would it be correct/easier to say

$\displaystyle [a,b]$ describes $\displaystyle \frac{1}{n}$ of this semicircle, where $\displaystyle n$ is obviously to be determined case wise.

So if we think about it we can parametrize our circle in terms

$\displaystyle x(\theta)=r\cos(\theta)$

and $\displaystyle y(\theta)=r\sin(\theta)$

So could we not say that

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

Which would be a much easier calculation. Is this correct in its assumption?

Once again, if this is soo apparently obvious sorry for posting, but if anyone could give me a feasible reason why this is not easier but at the same time using calculus, please do tell.

Mathstud