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What does the following represent: ∫−∞∞e−x2 dx=π\int_{-\infty}^\infty e^{-x^2} \,dx = \sqrt{\pi}


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The expression ∫−∞∞e−x2 dx=π\int_{-\infty}^\infty e^{-x^2} \,dx = \sqrt{\pi} represents a famous integral in mathematics, known as the Gaussian integral or the probability integral.

Here's what each part of the equation means:

  1. ∫−∞∞e−x2 dx\int_{-\infty}^\infty e^{-x^2} \,dx: This is the definite integral of the function f(x)=e−x2f(x) = e^{-x^2} over the entire real number line, from negative infinity to positive infinity. The function e−x2e^{-x^2} is known as the Gaussian function or the normal distribution, and it has a characteristic bell-shaped curve. The integral is used to find the area under this curve.

  2. =π= \sqrt{\pi}: This equation states that when you integrate the Gaussian function over the entire real line, the result is the square root of pi (approximately 1.77245). This constant value of π\sqrt{\pi} is a fundamental result in calculus and has applications in various fields, including probability theory, statistics, physics, and engineering.

In probability theory, this integral is significant because it is the integral of the probability density function of a standard normal distribution, which is widely used to model random variables in many natural and social phenomena. The value of π\sqrt{\pi} represents the total area under the Gaussian curve, indicating that the probability of a random variable following a standard normal distribution taking on any value between negative infinity and positive infinity is exactly π\sqrt{\pi}.

In summary, the equation ∫−∞∞e−x2 dx=π\int_{-\infty}^\infty e^{-x^2} \,dx = \sqrt{\pi} represents the calculation of the area under the Gaussian curve, and the result is a fundamental constant with important applications in mathematics and its related fields.


Evaluation

Evaluation ID
de34c3d8-aa68-4933-8c06-d05742e88e0b
Created
April 23, 2024

Evaluation Rating

Binary Pass/Fail
Pass
Score
100
Factual Consistency
C: the submitted answer contains all the same details as the expert answer
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