I am trying to understand the proof of the sphere area formula.
In my math book they use the formula $y = \sqrt{R^2 - x^2}$ $-R \leq x \leq R$
They rotate the function above around the x-axis and get:
$ A = 2\pi \int^R_{-R} y \sqrt{1+ (\frac{dy}{dx})^2} dx = 2\pi \int^R_{-R} \sqrt{R^2 - x^2} \sqrt{1+ \frac{x2}{R^2-x^2}} dx = 2\pi \int^R_{-R} \sqrt{R^2}dx = 4\pi R^2 $
I understand the development until this part:
$= 2\pi \int^R_{-R} \sqrt{R^2}dx = 4\pi R^2 $
Can someone please help me with this one (and how the calculation is made)? Please also explain your approach when solving it.
Thank you!