Using calculus, I attempted to find a formula for the surface area of an ellipsoid, which is a solid obtained by rotating the ellipse $\displaystyle\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ around the $x$-axis.
Link: Ellipsoid Surface Area ($C$ represents circumference, $A$ represents area)
Unfortunately, I don't think my formula $A_{ellipsoid} = \pi^2 ab$ is correct because in the case of a sphere with uniform radius $r$, my formula yields $\pi^2 r^2$ even though it is commonly known that a sphere's surface area is given by $4 \pi r^2$.
I would like to know what I did wrong in my derivation, so far I cannot spot any mistakes.
Surface area of revolution... uses this formula.
$2\pi\int_{-a}^{a} y\sqrt {1+(\frac {dy}{dx})^2} \ dx$
And you shouldn't need a trig substitution to evaluate it.