Find The Area of an Ellipse Using Calculus

Find the area of an ellipse using integrals and calculus.

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

ellipse used in problem

Solution to the problem:
The equation of the ellipse shown above may be written in the form
x 2 / a 2 + y 2 / b 2 = 1

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 2 / a 2 ]
The upper part of the ellipse (y positive) is given by
y = b √ [ 1 - x 2 / a 2 ]
We now use integrals to find the area of the upper right quarter of the ellipse as follows
(1 / 4) Area of ellipse = 0 a b √ [ 1 - x 2 / a 2 ] dx
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 = 0 π/2 a b ( √ [ 1 - sin2 t ] ) cos t dt
√ [ 1 - sin2 t ] = cos t since t varies from 0 to π/2, hence
(1 / 4) Area of ellipse = 0 π/2 a b cos2 t dt
Use the trigonometric identity cos2 t = ( cos 2t + 1 ) / 2 to linearize the integrand;
(1 / 4) Area of ellipse = 0 π/2 a b ( cos 2t + 1 ) / 2 dt
Evaluate the integral
(1 / 4) Area of ellipse = (1/2) b a [ (1/2) sin 2t + t ]0 π/2
= (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.