How to prove the inequality about the integral of $e^{-(x^2+y^2)}$ on square and disk with same area? I meet this exercise in double integral, let $D_r=\{(x,y)\in \mathbb{R}^2|x^2+y^2\leq \frac{4r^2}{\pi}\}, S_r=\{(x,y)\in\mathbb{R}^2|-r\leq x\leq r,-r\leq y\leq r\}$, which are disk and square centered at origin with the same area, I am asked to show $\iint_{D_r}e^{-(x^2+y^2)}dxdy> \iint_{S_r}e^{-(x^2+y^2)}dxdy$.
I can't find a method to compare these values, calculator shows that the difference  is small.
Thanks for any comment!
 A: Consider the value of $f(x,y) := e^{-(x^2+y^2)}$ on $D_r \setminus S_r$ and on $S_r \setminus D_r$. Since $D_r \setminus S_r \subseteq D_r$, we have $e^{-(x^2+y^2)} \geq e^{-4r^2/\pi}$ on $D_r \setminus S_r$. Since $S_r \setminus D_r \subseteq \mathbb R^2 \setminus D_r$, we have $e^{-(x^2+y^2)} \leq e^{-4r^2/\pi}$ on $S_r \setminus D_r$. Now observe that
\begin{align*}
&\iint_{D_r} f(x,y) \,dx \, dy - \iint_{S_r} f(x,y) \, dx \, dy \\
&= \left( \iint_{D_r \setminus S_r} f(x,y) \,dx \, dy + \iint_{D_r \cap S_r} f(x,y) \,dx \, dy \right) \\&\quad- \left( \iint_{S_r \setminus D_r} f(x,y) \,dx \, dy + \iint_{S_r \cap D_r} f(x,y) \,dx \, dy \right) \\
&= \underbrace{\iint_{D_r \setminus S_r} f(x,y) \, dx \, dy}_{\geq e^{-4r^2/\pi} \mu (D_r \setminus S_r)} - \underbrace{\iint_{S_r \setminus D_r} f(x,y) \, dx \, dy}_{\leq e^{-4r^2/\pi} \mu(S_r \setminus D_r)} \\
&\geq e^{-4r^2/\pi} \mu (D_r \Delta S_r) \\
&>0
\end{align*}
Here $\mu$ denotes the area and $D_r \Delta S_r = (D_r \setminus S_r) \cup (S_r \setminus D_r)$ denotes the symmetric difference. Since the difference between the two integrals is positive, we conclude that
$$\iint_{D_r} f(x,y) \,dx \, dy > \iint_{S_r} f(x,y) \,dx \, dy$$
