Find the double-integral $\iint_{D} x\sqrt{4x^2+xy}$ on $D=\lbrace (x,y) \in \mathbb{R^2}|x=a\cos(t),y=a\sin(2t) \rbrace$ and $x \geq 0$

Question

I am understant that the problem is finding the limits of the integration in the region $D$, so first i draw the region $D$ which is a kind of "infinity region", so and find the principal point where this region intersects the $x$-axis and $y$-axis and $x \geq 0$ and these are $(0,a),(0,0)$ so when i try say for example while $x\in [0,a]$ i cant find the two curves equation in which y belong. Any help and recomendation help me much. Sorry my English skills are low.

Answer 1

For your curve, known as member of family Lissajous curves, is possible to get representation $y=\pm2x\sqrt{1-\frac{x^2}{a^2}}$: second equation we can write as $y=2x \sin t$ and then solve $\sin t$ from 1-st equation.

If we understand your $D$ as area bounded by given curve with $x\geqslant 0$. Integral is $$\int\limits_{0}^{a}\int\limits_{-2x\sqrt{1-\frac{x^2}{a^2}}}^{2x\sqrt{1-\frac{x^2}{a^2}}}x\sqrt{4x^2+xy}\,dxdy$$