Find Limits of Integration in Polar Coordinates with Laplace Math Help

[vanceEE]

px = t
t = s^2

$$ I = \int_0^∞ e^{-s^2}ds$$
$$I*I = \int_0^∞ e^{-s^2}ds * \int_0^∞ e^{-u^2}du = \int_0^∞\int_0^∞ e^{-(s^2+u^2)}du ds$$
$$s = rsin\theta $$
$$u = rcos\theta $$
$$r = s^2 + u^2 $$
$$ I*I = \int_0^∞\int_\alpha^\beta e^-{r^2}rdrd\theta$$
How can I find my limits of integration in polar coordinates?

 

[pasmith]

px = t
t = s^2

$$ I = \int_0^∞ e^{-s^2}ds$$

What is your original integral? Is it
<br /> I = \int_0^\infty \frac{1}{\sqrt x}e^{-px}\,dx<br />
where the substitution px = s^2 is correct, yielding
<br /> I = \int_0^\infty \frac{1}{\sqrt x}e^{-px}\,dx = <br /> \int_0^\infty \frac{\sqrt {p}}{s} e^{-s^2} \frac{2s}{p}\,ds = <br /> \frac 2{\sqrt{p}} \int_0^\infty e^{-s^2}\,ds = \frac 1{\sqrt p} \int_{-\infty}^\infty e^{-s^2}\,ds.<br />

$$I*I = \int_0^∞ e^{-s^2}ds * \int_0^∞ e^{-u^2}du = \int_0^∞\int_0^∞ e^{-(s^2+u^2)}du ds$$
$$s = rsin\theta $$
$$u = rcos\theta $$
$$r = s^2 + u^2 $$
$$ I*I = \int_0^∞\int_\alpha^\beta e^-{r^2}rdrd\theta$$

You mean
<br /> I^2 = \int_\alpha^\beta \int_0^\infty e^{-r^2} r\,dr\,d\theta <br /> = \int_\alpha^\beta \left(\int_0^\infty e^{-r^2} r\,dr\right)\,d\theta<br />
not
<br /> \int_0^\infty \int_\alpha^\beta e^{-r^2} r\,dr\,d\theta <br /> = \int_0^\infty \left(\int_\alpha^\beta e^{-r^2} r\,dr\right)\,d\theta<br />

How can I find my limits of integration in polar coordinates?

You need both \sin \theta and \cos \theta to be positive. What does that give you?

 

[HallsofIvy]

Your two integrals, in x and y, were from 0 to infinity so the two integrals cover the first quadrant. In order to do that in polar coordinates you have to have r from 0 to infinity and \thetafrom 0 to \pi/2.