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CBSE Class 11-science Answered

PLEASE TELL ME HOW TO DERIVE THE FORMULA FOR AREA OF A CIRCLE USING BOTH DIFFERENTIAL AND INTEGRAL CALCULUS.

Asked by kandappan | 25 Sep, 2019, 10:12: AM

Expert Answer

By integration:-

Area of the circle is obtained by finding the area A under curve whose equation is y = (a² - x² )^1/2 , where a is radius of circle.

Area of circle is four times of A.

Let us consider a strip of width dx. Area of this strip is ( y dx ).

Area A is obtained by summing the area dA of such strip from x=0 to x=a.

Hence

begin mathsize 14px style A space equals space integral d A space equals space integral subscript 0 superscript a y d x space equals space integral subscript 0 superscript a square root of a squared minus x squared end root space d x end style

Above intergration is performed by using substitution x = a sinθ.

By integration we get, A = (π/4)a²

Hence area of circle = 4 A = πa² Sq. Units

-------------------------------------------------------------------------

Integration is nothing but continuous sum.

Hence area is considered as summation of small area elements that form the circle area and area is calculated by integration.

Differentiation is change in function for a small change of function variable.

It is not possible to find a method of finding area of circle by differentiation

Answered by Thiyagarajan K | 25 Sep, 2019, 12:24: PM

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