Find the area of this figure using polar coordinates (possible textbook mistake)

Question

I am required to find the area of this region using polar coordinates:

My setup is

$$ A = \frac{1}{2} \int_{0}^{\phi} \left[ R \sin(\theta) \right]^2 d\theta = \frac{R^2}{4} \int_{0}^{\phi} \left[ 1-\cos(2\theta) \right] = \frac{R^2}{4} \left[ \theta - \sin(2 \theta) \right] \bigg|_{0}^{\phi} = \frac{R^2 \phi}{4} - \frac{R^2 \sin(2 \phi)}{8} $$

The answer on the textbook is

$$ A = R^2\phi - \frac{R^2\sin(2\phi)}{2} $$

Am I making a mistake somewhere?

Answer 1

It is easy to check that for $\varphi = \pi,\pi/2 $ your answer tallies as a semi-circle and full circle of diameter $R$. The text book is wrong.

His error is because of assuming the label $R$ for diameter as radius.