polar coordinates integral equation 
Let $\phi$ be a k- times differentiable function. Prove that the following holds:
$\int_{t_o}^{t_1}(\alpha_1(s)\dot \alpha_2(s)- \alpha_2(s)\dot \alpha_1(s))ds=\phi(t_1)-\phi(t_o)$ where $\alpha(s)=(\cos(\phi(s)),\sin(\phi(s)))$

I guess I would only need to show that $\dot \phi(s)=\alpha_1(s)\dot \alpha_2(s)-\alpha_2(s)\dot \alpha_1(s)$ but I dont know how.
Would appreciate help/hints

$$\alpha=(\cos\phi,\,\sin\phi)\implies\dot\alpha=(-\dot\phi\sin\phi,\,\dot\phi\cos\phi)$$
and thus (with the expression already corrected) 
$$\alpha_1\dot\alpha_2-\dot\alpha_1\alpha_2=\dot\phi\cos^2\phi+\dot\phi\sin^2\phi=\dot\phi$$
