I will refer to the curves as r(t) (where t - polar angle) to avoid confusion with Cartesian coordinates. I.e.
circle: r=3sin(t), cardioid r=1+sin(t)
Area in polar coordinates
1/2∫r2(t)dt
Draw simple chart
Region area
A=1/2∫t1t2(3sin(t))2−1/2∫t1t2(1+sin(t))2dt=∫t1t24sin2(t)−sin(t)−1/2dt
Find integration bounds as intersections of curves
3sin(t)=1+sin(t),0≤t≤2πsin(t)=1/2t1=π/6,t2=5π/6
Integrate
A=∫π/65π/64sin2(t)−sin(t)dt−1/2dt=[2t−sin(2t)+cos(t)−1/2t]π/65π/6=3/2(5π/6−π/6)−(sin(5π/3)−sin(π/3))+(cos(5π/6)−cos(π/6))=π+2sin(π/3)−2cos(π/6)=π+23/2−23/2=π≈3.1416
Answer: 3.1416
Comments
Leave a comment