Finding area boundes by polar curves

Question

Find the area bounded by $r \le 4sin(t), \frac{\pi}{3} \le t$

Well, when I graph it with Geogebra I get an entire circunference, but what values should I plot for the definite integral? Would it be from $\frac{\pi}{3}$ to $2 \pi$?

Answer 1

The range of $t$ for the definite integral should be $0<t<\pi/3$. So, the area integral is,

$$\int_0^{\pi/3}\int_0^{4\sin (t)}rdrdt=8\int_0^{\pi/3}\sin^2(t)dt=\frac{4}{3}\pi-\sqrt{3}$$