Given $\displaystyle 0< a \leq b,$ define $\displaystyle \underline{\psi}:[0, 2 + \infty) \times [0, 2 \pi] \rightarrow \mathbb{R}^2$ by $\displaystyle \underline{\psi}(r, \theta):=(arcos(\theta),br sin(\theta)).$

Given $\displaystyle 0 \leq \theta_1< \theta_2 \leq 2 \pi$, calculate the area of the sector of the ellipse $\displaystyle \left( \frac{x^2}{a^2} \right)+\left( \frac{y^2}{b^2} \right)=1$ defined by $\displaystyle \{ \underline{\psi}(r, \theta)|0 \leq r \leq 1, \theta_1 \leq \theta \leq \theta_2 \}$.