Question

Let $s$ be the set of points with polar coordinates $(r,\theta)$ such that $ 2 \le r \le 6$ and $\frac{\pi}{3} \le \theta \le \frac{5 \pi}{6} $. find the area of $s$.

Answer 1

The area defined by
2 ≤ r ≤ 6 and π/3 ≤ θ ≤ (5π)/6 is shown in the figure below.

An element of area is
dA = r dr dθ

Therefore the total area is

`A=\int _{ ( \pi )/(3)} ^{ (5 \pi )/(6) } \,d\theta \int_(2)^(6) \,rdr \\ A= [ (5 \pi )/(6)- ( \pi )/(3)]*[ (6^(2))/(2) - (2^(2))/(2)] = 8 \pi`

Answer: 8π