Find The Area of Circle Using Integrals in Calculus

Find the area of a circle of radius a using integrals.

Problem : Find the area of a circle with radius a.

Solution to the problem:

The equation of the circle shown above is given by

x^{ 2} + y^{ 2} = a^{ 2}

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

Solve the above equation for y

y = + or - √[ a^{ 2} - x^{ 2} ]

The equation of the upper semi circle (y positive) is given by

y = √[ a^{ 2} - x^{ 2} ]

= a √ [ 1 - x^{ 2} / a^{ 2} ]

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

(1 / 4) Area of cirle = _{0}^{a} a √ [ 1 - x^{ 2} / a^{ 2} ] dx

Let us substitute x / a by sin t so that sin t = x / a and dx = a cos t dt and the area is given by

(1 / 4) Area of cirle = _{0}^{pi/2} a^{ 2} ( √ [ 1 - sin^{2} t ] ) cos t dt

We now use the trigonometric identity

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

(1 / 4) Area of circle = _{0}^{pi/2} a^{ 2} cos^{2} t dt

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

(1 / 4) Area of circle = _{0}^{pi/2} a^{ 2} ( cos 2t + 1 ) / 2 dt

Evaluate the integral

(1 / 4) Area of circle = (1/2) a^{ 2} [ (1/2) sin 2t + t ]_{0}^{pi/2}

= (1/4) pi a^{ 2}

The total area of the circle is obtained by a multiplication by 4

Area of circle = 4 * (1/4) pi a^{ 2} = pi a^{ 2}
More references on
integrals and their applications in calculus.