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*a*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.

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To start with, we recognise that the formula for one quarter of an ellipse is
##y = b*sqrt((1-x^2)/a^2)##
This quarter-ellipse is “centred” at ##(0,0)##. Its area is
##A = int_0^a(b*sqrt((1-x^2)/a^2))dx##
So, naturally, the total area of the ellipse is
##A = 4int_0^a(b*sqrt((1-x^2)/a^2))dx##.

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