If we consider the equation of a circle:
$$x^2+y^2=R^2$$
Then I propose that the volume of a sphere of radius $R$ is given by the twice the summation of the circumferences of the circles between the origin and $x=R$ along the x axis, each circle having a radius equal to the value of y at that point in x.
Since
$$y={\sqrt{R^2-x^2}}$$
I derived the formula:
$$SA = 2\int^R_0{{\sqrt{R^2-x^2}}}dx$$
However, evaluating this and using integration by substitution (using $x=R\sin(u)$ to find the integral, I obtained:
$$SA=2\pi R^2\left[\frac{\sin(2u)}{2}+\frac{u}{2}\right]^{\pi /2}_0$$
I have checked this multiple times and I can't seem to see what the problem is. If the problem is with the original proposition, please could you explain why the proposition is incorrect.