# 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