# How do you find the area of an ellipse using integrals?

No matter which way you use integrals, the solution will always come out to be $\pi \cdot a \cdot b$, where a and b are the semi-major axis and semi-minor axis. However, if you insist on using integrals, a good way to start is to split the ellipse into four quarters, find the area of one quarter, and multiply by four.
$y = b \cdot \sqrt{\frac{1 - {x}^{2}}{a} ^ 2}$
This quarter-ellipse is "centred" at $\left(0 , 0\right)$. Its area is
$A = {\int}_{0}^{a} \left(b \cdot \sqrt{\frac{1 - {x}^{2}}{a} ^ 2}\right) \mathrm{dx}$
$A = 4 {\int}_{0}^{a} \left(b \cdot \sqrt{\frac{1 - {x}^{2}}{a} ^ 2}\right) \mathrm{dx}$.