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.



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 = + or - 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 =
0a 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 =
0pi/2 a b ( √ [ 1 - sin2 t ] ) cos t dt

√ [ 1 - sin
2 t ] = cos t since t varies from 0 to pi/2, hence

(1 / 4) Area of ellipse =
0pi/2 a b cos2 t dt

Use the trigonometric identity cos
2 t = ( cos 2t + 1 ) / 2 to linearize the integrand;

(1 / 4) Area of ellipse =
0pi/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 ]
0pi/2



= (1/4) pi a b

Obtain the total area of the ellipse by multiplying by 4

Area of ellipse = 4 * (1/4) pi a b = pi a b

More references on integrals and their applications in calculus.