Question #3fa8f
2 Answers
see explanation
Explanation:
consider circle
divide the circle in four quadrants, so area of circle=4
now, area of quadrant=
Consider a circle of radius
Method 1 - Polar Area Formula
We will calculate the area using Polar Coordinates. Polar area is given by the formula;
A = int_alpha^beta \ 1/2 r^2 \ d theta
For our circle of radius
Hence;
A = int_0^(2pi) 1/2 a^2 \ d theta
\ \ \ = 1/2 \ a^2 \ int_0^(2pi) \ d theta
\ \ \ = 1/2 \ a^2 \ [theta]_0^(2pi)
\ \ \ = 1/2 \ a^2 \ (2pi-0)
\ \ \ = a^2 pi \ \ QED
Method 2 - Double Integral
We could also consider a double integral, Suppose
r would be a ray varying from0 toa ;
theta would vary from0 to2pi
The area,
A = int int _R \ d A
\ \ \ = int_0^(2p) \ int_0^a \ r \ dr \ d theta
\ \ \ = int_0^(2pi) [ 1/2 r^2]_0^a \ d theta
\ \ \ = int_0^(2pi) ( 1/2 a^2 - 0) \ d theta
\ \ \ = 1/2 a^2 \ int_0^(2pi) \ d theta
\ \ \ = 1/2 a^2 \ [ theta]_0^(2pi) \ d theta
\ \ \ = 1/2 a^2 \ (2pi-0)
\ \ \ = a^2pi \ \ QED