How to prove the inequality $\iint_{x^2+y^2 \leq 1} e^{x^2+y^2}dxdy \leq (\int_{\sqrt{\pi}/2}^{\sqrt{\pi}/2} e^{x^2}dx)^2$
Prove $\iint_{x^{2}+y^{2} \leq 1} e^{x^{2}+y^{2}} d x d y \leq\left[\int_{-\frac{\sqrt{\pi}}{2}}^{\frac{\sqrt{\pi}}{2}} e^{x^{2}} d x\right]^{2} $