How do you prove area of a circle?
Integrate an incremental area
dArea = r dr dTheta from r=0 to R, and Theta from zero to 2PI radians.
Area= INT INT dTheta rdr dTheta
Area= 2PI (1/2 R^2)= PI R^2
To prove the area of a circle, we can use integration.
First, we need to consider an infinitesimally small area element on the circle. This area element can be visualized as a small rectangle with one side of length dr, representing the infinitesimally small change in the radius, and the other side of length r*dθ, representing the arc length corresponding to the infinitesimally small change in angle θ.
Therefore, the infinitesimal area dA can be given as dA = r dr dθ.
Next, we integrate this infinitesimal area over the entire circle. The integration is performed with respect to both the radius and the angle.
The limits of integration for the radius are from 0 to R, covering the entire range of the radius of the circle. The limits of integration for the angle are from 0 to 2π radians, covering a complete revolution around the circle.
So, the double integral to evaluate the area A of the circle becomes:
A = ∬ dA = ∫∫ r dr dθ
Integrating with respect to r, we get:
A = ∫ (r^2 / 2) dθ
Integrating with respect to θ, we get:
A = (1/2) r^2 θ | from 0 to 2π
Evaluating the limits, we have:
A = (1/2) R^2 (2π)
Simplifying, we get:
A = π R^2
So, the proof shows that the area of a circle is equal to π times the square of its radius.