I've been interested in properly calculating a Jacobian Integral (that is, an integral in which a change of variables occurs). I'm sure that, by one's reading this, you've all probably already heard of how Jacobian Integration was used to calculate $$\int_{-\infty}^{\infty}e^{-x^2}dx = \sqrt{\pi}$$. In that example, switching to polar coordinates yields the Jacobian Determinant $r$, such that the integrand can be changed to $re^{-r^2}$. And, to access all valid $\theta$ and $r$ as was done with $x$ and $y$ in the original double integral, the limits become $\theta\in[0,2\pi)$ and $r\in[0,\infty)$.
All of this makes sense to me. However, say I wanted to integrate similarly; more specifically, $$\int_{-1}^{1}e^{-x^2}dx$$ (I actually want to do this for a different function, but I figured I would keep it to this good example integral). Obviously, it is still clear to me how to change to polar coordinates, keeping the same Jacobian determinant, but how should I map the limits of integration to polar form?
Intuition tells me that I should utilize polar coordinates' definitions, $x=r\cos\theta$, $y=r\sin\theta$, and $r^2 = x^2+y^2$. This would imply that our limits should be over $r\in[0,1],\ \theta\in[-\frac\pi4,\frac\pi4]$. However, I'm uncertain and might be wrong. Am I headed in the right direction? What should I be doing? Are these limits possible to integrate analytically?
Thanks in advance!
It's not going to work. Your $r$ limits are way more complicated. You should get $$8 \int_0^{\pi/4}\int_0^{\sec\theta} re^{-r^2}\,dr\,d\theta.$$ Good luck!!