Math 2.0

example

Integral of exp(-x) over the half line

Example

The improper integral of e−xe^{-x} over [0,∞)[0, \infty) converges, and ∫0∞e−x dx=1.\int_0^\infty e^{-x} \, dx = 1. For b>0b > 0, the exponential formula with c=−1c = -1 gives ∫0be−x dx=1−e−b\int_0^b e^{-x} \, dx = 1 - e^{-b}. This tends to 11 as b→∞b \to \infty.

Remarks

It is the case s=1s = 1 of the gamma function: Γ(1)=1\Gamma(1) = 1. Mathlib states it as integral_exp_neg_Ioi_zero.

Explanations

L1 human

The region under e−xe^{-x} is infinitely long, but its area is finite. The area is exactly 11.

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  55. Integral of exp(-x) over the half line example

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Sources

  1. Web https://en.wikipedia.org/wiki/Improper_integral, Convergence of the integral

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