Consider the following curve in polar coordinates.
\[ r = 2 - 2\sin\theta \]
Which one of the following is the area enclosed by the curve for $0 \le \theta \le 2\pi$?

Question

For a first-order irreversible reaction carried out in a batch reactor, the rate of reaction is proportional to:

Answer 1

Rotate to $r=2(1-\cos\phi)$, a standard cardioid with a=2. $\int_0^{2\pi}r^2d\phi=3\pi a^2=12\pi$. $A=\frac12(12\pi)=6\pi$.\[ A=6\pi \]

Consider the following curve in polar coordinates.
\[ r = 2 - 2\sin\theta \]
Which one of the following is the area enclosed by the curve for \(0 \le \theta \le 2\pi\)?