Answer to Question #336502 in Calculus for PEAC_BOY

Question #336502

1.Find the area of the region that lies inside the circle r =3sinx and outside the cardioid

r = 1+sinx



1
Expert's answer
2022-05-04T11:46:17-0400

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

1/2r2(t)dt1/2\int r^2(t)dt


Draw simple chart




Region area

A=1/2t1t2(3sin(t))21/2t1t2(1+sin(t))2dt=t1t24sin2(t)sin(t)1/2dtA = 1/2\int_{t_1}^{t_2} (3\sin(t))^2 - 1/2\int_{t_1}^{t_2} (1+\sin(t))^2dt\\ = \int_{t_1}^{t_2} 4\sin^2(t) - \sin(t) -1/2 dt


Find integration bounds as intersections of curves

3sin(t)=1+sin(t),0t2πsin(t)=1/2t1=π/6,t2=5π/63\sin(t) = 1+ \sin(t), 0 \le t \le 2\pi\\ \sin(t) = 1/2\\ t_1 = \pi/6, t_2=5\pi/6


Integrate

A=π/65π/64sin2(t)sin(t)dt1/2dt=[2tsin(2t)+cos(t)1/2t]π/65π/6=3/2(5π/6π/6)(sin(5π/3)sin(π/3))+(cos(5π/6)cos(π/6))=π+2sin(π/3)2cos(π/6)=π+23/223/2=π3.1416A= \int_{\pi/6}^{5\pi/6}4\sin^2(t) - sin(t)dt - 1/2dt\\ = \Big[2t -\sin(2t) +\cos(t) - 1/2t\Big]_{\pi/6}^{5\pi/6}\\ = 3/2(5\pi/6 -\pi/6) - (\sin(5\pi/3) - \sin(\pi/3)) + (\cos(5\pi/6) - \cos(\pi/6))\\ = \pi +2\sin(\pi/3) - 2\cos(\pi/6)\\ = \pi + 2\sqrt{3}/2 - 2\sqrt{3}/2\\ = \pi \approx 3.1416



Answer:  3.1416


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