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 from#0# to#a# ;
#theta# would vary from#0# to#2pi #
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