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
[tex]A=\int _{ \frac{ \pi }{3}} ^{ \frac{5 \pi }{6} } \,d\theta \int_{2}^{6} \,rdr \\
A= [ \frac{5 \pi }{6}- \frac{ \pi }{3}]*[ \frac{6^{2}}{2} - \frac{2^{2}}{2}] = 8 \pi [/tex]
Answer: 8π
