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.

circle used in the calculation of integral


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 =
0a 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 =
0pi/2 a 2 ( √ [ 1 - sin2 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 =
0pi/2 a 2 cos2 t dt

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

(1 / 4) Area of circle =
0pi/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 ]0pi/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.

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