Find the area of an ellipse using integrals and calculus.

__Problem :__ Find the area of an ellipse with half axes a and b.

The equation of the ellipse shown above may be written in the form

Since the ellipse is symmetric with respect to the x and y axes, we can find the area of one quarter and multiply by 4 in order to obtain the total area.

Solve the above equation for y

y = ± b √ [ 1 - x

The upper part of the ellipse (y positive) is given by

y = b √ [ 1 - x

We now use integrals to find the area of the upper right quarter of the ellipse as follows

(1 / 4) Area of ellipse =

We now make the substitution sin t = x / a so that dx = a cos t dt and the area is given by

(1 / 4) Area of ellipse =

√ [ 1 - sin

(1 / 4) Area of ellipse =

Use the trigonometric identity cos

(1 / 4) Area of ellipse =

Evaluate the integral

(1 / 4) Area of ellipse = (1/2) b a [ (1/2) sin 2t + t ]

= (1/4) π a b

Obtain the total area of the ellipse by multiplying by 4

Area of ellipse = 4 * (1/4) π a b = π a b More references on integrals and their applications in calculus.