This problem is related to question i asked before. Link
Problem
Formula for area was proven in the other post but i still have some problem with integration (with the arithmetic itself).
$$ A=2\pi \int_{a}^{b} |f(x)| \sqrt{1+f'(x)^2}dx $$
This is formula for area of cap of a sphere if revovling function is known. $f(x)$ is the revolving function in this formula.
Our revolving function is:
$$ f(x)=\sqrt{R^2-x^2} $$
$$ A=2\pi \int_{a}^{b} |f(x)| \sqrt{1+f'(x)^2}dx $$
We wanted to know area when $$ (R-h) \le x \le R $$
Now simply putting all these together we get.
$$ A=2\pi \int_{R-h}^{R} |\sqrt{R^2-x^2}| \sqrt{1+(-\frac{x}{\sqrt{R^2-x^2}})^2}dx $$
Now when i try to compute indefinite integral i don't get so far. I have no clue how to integrate something like this. $$ A=2\pi \int |\sqrt{R^2-x^2}| \sqrt{1+(-\frac{x}{\sqrt{R^2-x^2}})^2}dx $$ $$ A=2\pi \int |({R^2-x^2})^\frac{1}{2}| ({1+(-\frac{x}{R^2-x^2})^2})^{\frac{1}{2}}dx $$
Well i tried this to integrate this with wolframalpha and it returns following:
Indefinite integral:
No result found in terms of standard mathematical functions.
However when you put in the integration limits in wolframalpha it gives correct answer. Problem is i don't understand why ?