I am trying to derive the formula for the area of a sphere using integration. It is coming out as $ \pi^2R^2 $ instead of $4\pi R^2$.
This is what I am doing :-
I am approximating the area of the sphere of radius R (kept at origin) using the Curved Surface Area of infinite infinitesimal cylinders along the X axis.
Now each infinitesimally small cylinder's Curved Surface Area is $ 2π f(x) dx $.
Therefore, Area of the sphere is :-
$$ \int_{-R}^R 2\pi f(x) dx \\ = 2\pi\int_{-R}^R \sqrt{R^2 - x^2} dx \\ \text{Put } x = R \sin \theta \text{, we get} \\ \text{Area} = 2\pi \int_{-\frac{\pi}{2}}^\frac{\pi}{2} R \cos \theta. R \cos \theta d\theta \\ = 2 \pi R^2 \int_{-\frac{\pi}{2}}^\frac{\pi}{2} \cos^2 \theta d \theta \\ = 2 \pi R^2 \int_{-\frac{\pi}{2}}^\frac{\pi}{2} \frac{1 + \cos2\theta}{2} d\theta \\ = \pi R^2 \left [\theta + \frac{\sin2\theta}{2} \right ]_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \\ = \pi^2 R^2 $$
What am I doing wrong?