I need to find the surface area of an ellipsoid using the equation of an ellipse. I believe my calculations are correct but the formulas I meet on the Internet are complex and have $\arcsin$ or $\arctan$ in the equation, which makes me believe that there is something wrong in my calculations.
Can anyone help me if there is something wrong? The integration limits are $0$ to $a$ to calculate half of the area but multiplied by $2$ to calculate the whole area.
\begin{align*} y(x) &= \sqrt{b^2 a^2-x^2} \\ y'(x) &= -\frac{x}{\sqrt{b^2a^2-x^2}} \\ S &= 2\pi \int_a^b f(x)\sqrt{1 + y'^2} dx \\ &= 4\pi\int_0^a \sqrt{b^2a^2-x^2} \sqrt{1 + \left( \frac{-x}{\sqrt{b^2a^2-x^2}} \right)^2} dx \\ &= 4\pi\int_0^a \sqrt{b^2a^2-x^2} \sqrt{1 + \left( \frac{x^2}{b^2a^2-x^2} \right)} dx \\ &= 4\pi\int_0^a \sqrt{b^2a^2-x^2} \sqrt{\frac{b^2a^2}{b^2a^2-x^2}}dx \\ &= 4\pi\int_0^a \sqrt{b^2a^2} dx \\ &= 4\pi\sqrt{b^2a^2} \left[ x \right]_0^a \\ &= 4\pi a\sqrt{b^2a^2} \end{align*}