Can't we use the following technique for N=1:

Set \(\displaystyle I=\int_{0}^b e^{x^2}dx\). Then \(\displaystyle I^2=\int_{0}^b \int_{0}^b e^{x^2+y^2}dx dy\). Now we can at least say that \(\displaystyle \int_{0}^{2\pi} \int_{0}^{b/2} e^{r^2}r dr d\theta\leq I^2\leq \int_{0}^{2\pi} \int_{0}^b e^{r^2}r dr d\theta\). Furthermore, the upper and lower bounds can be computed explicitly.

Isn't this correct firstly? and if so, can't we extend this to more dimensions?

Thanks.