Area shape calculating

Question

Can't find the area of the figure bounded by the curve in polar coordinates $$\phi=r\arctan(r), \phi=0, \phi=\frac{\pi}{\sqrt 3}.$$ I tried use the formula $$S=\frac 12\int_{0}^{\frac{\pi}{\sqrt 3}}(r^2(\phi))d\phi$$ but can't to find $r(\phi)$.

Answer 1

You don't need to get $r(\phi)$. Since $$d\phi=\left(\arctan r+\frac{r}{1+r^2}\right)dr$$ you can have $$\begin{align}\int_{0}^{\frac{\pi}{\sqrt 3}}\frac 12r^2d\phi&=\int_{0}^{\sqrt 3}\frac 12r^2\left(\arctan r+\frac{r}{1+r^2}\right)dr\\&=\frac 12\int_{0}^{\sqrt 3}r^2\arctan rdr+\frac 12\int_{0}^{\sqrt 3}\frac{r^3}{1+r^2}dr\end{align}$$

For the former, this question will help.