Question

Find the area of the ellipse with integration and examples

Answer 1

Answer:

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 = ~+mn~ 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 = 0π/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 = 0π/2 a b cos2 t dt

Use the trigonometric identity cos2 t = ( cos 2t + 1 ) / 2 to linearise the integrand;

(1 / 4) Area of ellipse = 0π/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 ]0π/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.

Step-by-step explanation: