This page titled 10.2: B- Evaluating the Gaussian Integral is shared under a CC BY-SA license and was authored, remixed, and/or curated by Daniel F. Styer.

Best buy laser pointersgreen

The LibreTexts libraries are Powered by NICE CXone Expert and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Legal. Accessibility Statement For more information contact us at info@libretexts.org.

called the Gaussian integral, does not fall to any of the methods of attack that you learned in elementary calculus. But it can be evaluated quite simply using the following trick.

Best buy laser pointersfor sale

At the last step we have written A2 as a two-variable integral over the entire plane. This seems perverse, because most of the times we work hard to reduce two-dimensional integrals to one-dimensional integrals, whereas here we are going in reverse. But look at the integrand again. When regarded as an integral on the plane, it is clear that we can regard x2 + y2 as just r2, and this suggests we should convert the integral from Cartesian (x, y) to polar (r, θ) coordinates:

\[ A^{2}=\int_{-\infty}^{+\infty} e^{-x^{2}} d x \int_{-\infty}^{+\infty} e^{-y^{2}} d y=\int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} d x d y e^{-\left(x^{2}+y^{2}\right)}.\]