Question

I. Find the area enclosed by each of the following.

1. r = 4 Sin θ

2. r = 2 - cos θ

II. Find the common area enclosed by the following pairs of curves.

1. r = 3 Cos θ

2. r = 1 + Cos θ

III. Find the area which is inside of the curve r = 5 Sin θ and the outside of the curve r = 2 + Sin θ.

Answer 1

Question 1

$A=\int _a^b|f\left(x\right)-g\left(x\right)|dx$

$=\int _0^{2\pi }\left|4\sin \left(θ\right)-\left(2-\cos \left(θ\right)\right)\right|dθ=18.47376$

Question 2

$=\int _0^{2\pi }\left|3\cos \left(θ\right)-\left(1+\cos \left(θ\right)\right)\right|dθ=-2\pi +\frac{8\pi }{3}+4\sqrt{3}$

Question 3

$=\int _0^{2\pi }\left|5\sin \left(θ\right)-\left(2+\sin \left(θ\right)\right)\right|dθ=4\pi +8\sqrt{3}-\frac{8\pi }{3}$