<s>[INST] Evaluate $\iint_{D}\sqrt{1-\frac{x^2}{a^2}-\frac{y^2}{b^2}}\text{d}x\text d y$ 
Evaluate using polar coordinates:$$\iint_{D}\sqrt{1-\frac{x^2}{a^2}-\frac{y^2}{b^2}}\text{d}x \, \text d y\qquad D=\bigg\{\frac{x^2}{a^2}+\frac{y^2}{b^2}\leqslant 1\bigg\}$$

What I did:
I found that the Jacobian matrix equal to $a\cdot b\cdot r$ I am sure about that and then I tried this:
$$\int_0^{2\pi}\int_0^1 \sqrt{1-x^2/a^2-y^2/b^2} \cdot (abr) \, \text{d} r \, \text d\theta$$ which leads to wrong answer,
a small bird whisper in my ear that I should get this:
$$\int_0^{2\pi}\int_0^1 \left(\sqrt{1-r^2}abr\right) \, \text{d}r \, \text d\theta$$ 
I wanted to tell her that I don't understed how to come to this result but she fly away, can you help me with that?
 [/INST] If the change is $x = a r\cos\theta$, $y = b r\sin\theta$:
$$\frac{x^2}{a^2} + \frac{y^2}{b^2} = r^2\cos^2\theta + r^2\sin^2\theta = r^2.$$
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