How to prove the value of this integral?

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Discussion Overview

The discussion centers around the evaluation of the integral \(\int^{\infty}_{-\infty} e^{-x^{2}}dx\). Participants explore various methods for proving its value, including the use of double integrals and polar coordinates, while also considering the possibility of using single-variable calculus techniques.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • One participant states the integral evaluates to \(\frac{\sqrt{\pi}}{2}\).
  • Another suggests considering the double integral \(\int_{-\infty}^{\infty} \int_{-\infty}^\infty e^{-x^2-y^2}\, dx \, dy\) as a hint for the proof.
  • A participant notes that by symmetry, the double integral can be expressed as \(4\int_0^\infty\int_0^\infty e^{-x^2- y^2} dx dy\) and mentions the potential to convert this into a more manageable form using polar coordinates.
  • One participant expresses uncertainty about using double integrals and questions whether the result can be proven using only single-variable calculus, mentioning attempts with integration by parts and limit of sums.
  • Another participant echoes the previous point about the possibility of using symmetry and suitable manipulations to evaluate the integral, indicating challenges faced in direct summation.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the method of proof, with some advocating for double integrals and others questioning the feasibility of using single-variable calculus techniques.

Contextual Notes

Some participants express limitations in their understanding of double integrals, which may affect their ability to engage with the proposed methods fully. There is also a mention of failed attempts at integration by parts and direct summation, indicating unresolved mathematical steps in their reasoning.

suchith
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[itex]\int^{∞}_{-∞} e^{-x^{2}}dx[/itex] = [itex]\frac{\sqrt{\pi}}{2}[/itex]
 
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Ah, that one. Hint: consider
##\int_{-\infty}^{\infty} \int_{-\infty}^\infty e^{-x^2-y^2}\, dx \, dy##
 
That should be found in pretty much any Calculus text. Look at the integral pwsnafu suggests. Note that, by symmetry, that is [itex]4\int_0^\infty\int_0^\infty e^{-x^2- y^2} dx dy[/itex], over the first quadrant. That can be converted into a "doable" integral by changing to polar coordinates.
 
HallsofIvy said:
That should be found in pretty much any Calculus text. Look at the integral pwsnafu suggests. Note that, by symmetry, that is [itex]4\int_0^\infty\int_0^\infty e^{-x^2- y^2} dx dy[/itex], over the first quadrant. That can be converted into a "doable" integral by changing to polar coordinates.
I don't know Double integrals. Is it possible to prove the result only using Single variable calculus? At first I tried Integration by parts, but I failed :(

Isn't it possible to integrate using limit of sums and symmetry with suitable manipulations? I tried to sum directly but failed :(
 

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