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

Answer 1

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.

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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Answer 2

Axel Alvarez

To find the area of an ellipse using integrals, you can use the formula:

[ A = 4a\int_{0}^{b}\sqrt{1-\frac{x^2}{b^2}}dx ]

where ( a ) is the length of the semi-major axis and ( b ) is the length of the semi-minor axis of the ellipse.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.