How to find the area bounded by the curves $x=a\cos t$ and $y=b\sin t$ in the first quadrant?

Question

I know the given equations are the parametric equation of an ellipse . The curve meet the x axis at $(a,0)$ in the first quadrant .
Now I do this $\int_{0}^{a} y dx$
My book has the following step which I am unable to understand
$\int_{\frac{\pi}{2}}^{0} (b\sin t)(-a\cos t)dt $
Please explain this . Thank you !

Answer 1

$$x=a\cos{t}\Longrightarrow dx=-a\sin{t}dt$$ $$x=0\Longrightarrow t=\dfrac{\pi}{2}$$ $$x=a\Longrightarrow t=0$$ Thus you can rewrite your integral as: $$\int_0^ay\ dx=\int_{\pi/2}^0(b\sin{t})(-a\sin{t})dt=\ldots=\dfrac{1}{4}\pi ab=\dfrac{1}{4}\mathcal{A}_{\text{ ellipse}}$$