Consider a thin spherical shell. Lets look at 1 small element of the shell, which is like a longitude of the shell, and let it subtend an angle of $d\theta$ at the centre of the shell.
It has a surface area $dA=2\pi$r*rd$\theta$.
So we can consider many such small elements, each like a longitude of the shell, and each passing through the same 2 poles of the sphere.
Thus we can integrate this expression for surface area, taking limits as 0 to $\pi$, and this will cover the entire area of the sphere. So,
$$A=2\pi r^2\int_0^\pi\,d\theta$$
i.e. $A=2\pi^2$r$^2$
Where have I gone wrong?
Here in the image, the green part represents one element, not red.
Consider the violet element of area in your picture, comprised between $\theta$, $\theta+d\theta$ in longitude, and between $\phi$, $\phi+d\phi$ in latitude. Its area is $$ dA'=r\cos\phi\,d\theta\cdot r\,d\phi. $$ Integrate this on $\phi$ for $-\pi/2\le\phi\le\pi/2$ to get the area of the whole slice as $dA=2r^2\,d\theta$.
Integrating again on $\theta$ for $0\le\theta\le2\pi$ one gets the area of the sphere.