Finding area between polar curves

Question

Use double integral in polar coordinates to find the area of the region inside the circle x² + y² = 4 but outside the circle (x − 1)² + y² = 1.

Answer 1

To see the area it is recommended to draw those two circles.

The second circle is located inside the first one. So the area would be the area of the first circle minus the area of the second one.

 

x²+y²=4

Convert to polar coordinates: x=rcos(φ), y=rcos(φ), 0≤r≤2,0≤φ≤2π

Area1=∫∫rdrdφ=2πr²/2|₀²=4π  (first circle)

(x-1)²+y²=1
Convert to polar coordinates: x-1=rcos(φ), y=rcos(φ), 0≤r≤1,0≤φ≤2π
Area1=∫∫rdrdφ=2πr2/2|02=π (first circle)

 

Area2=∫∫rdrdφ=π

 

Area=Area1-Area2=3π