One can easily find the integral $\int_{0}^{\infty}\exp(-x)dx$. It is equal to 1. But is there a way to understand this geometrically without integration?

If i rotate the picture i see that $\int_{0}^{\infty}\exp(-x)dx=-\int_{0}^{1}\ln(t)dt$. Maybe there is some property of exp or log which allows to avoid integration?


I would like to accept the Mamikon's method pointed out by Jim Belk. But it is impossible to accept comments... So I accept the second best.

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    $\begingroup$ Why do you want to avoid integration? The exponential is one of the simplest functions to integrate. If you want to make things a little bit simpler, change the variable $x=\ln t$. $\endgroup$ – Beni Bogosel May 30 '11 at 8:25
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    $\begingroup$ See math.stackexchange.com/questions/3444/… for a generalized variant of this question. $\endgroup$ – Jonas Meyer May 30 '11 at 8:30
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    $\begingroup$ See its.caltech.edu/~mamikon/VisualCalc.html for a geometric explanation of this integral. $\endgroup$ – Jim Belk May 30 '11 at 8:48
  • $\begingroup$ Wow! Thanks for the links! $\endgroup$ – Roah May 30 '11 at 9:23
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    $\begingroup$ If you believe that $\int_0^{\infty} a^{-x} \, dx$ is finite for any $a$, then you can regard this integral as a definition of $e$ (that is, it's the unique value of $a$ that makes the integral equal to $1$). $\endgroup$ – Qiaochu Yuan May 30 '11 at 11:27

The following argument uses only basic properties of the exponential function and the integral, but not the fundamental theorem of calculus:

Let $\int_0^\infty e^{-x}dx=:I$. By looking at a figure we see that for any $c>0$ we have $$I=\int_0^c e^{-x}dx +\int_c^\infty e^{-x}dx= \int_0^c e^{-x}dx + e^{-c} I$$ or $$(1-e^{-c}) I =\int_0^c e^{-x}dx\ .$$ Using $e^{-c}\leq e^{-x}\leq1 \ \ (0\leq x\leq c)$ we conclude that $$c \ e^{-c} \leq (1-e^{-c}) I \leq c\ .$$ Now divide by $1-e^{-c}$ and let $c\to 0+$ to get the desired result.

  • $\begingroup$ This still uses integration... $\endgroup$ – Najib Idrissi May 30 '11 at 10:32
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    $\begingroup$ @zulon: Where ? $\endgroup$ – joriki May 30 '11 at 11:16
  • $\begingroup$ Oops. Nowhere. I misread what you wrote. Sorry :/. But there's still the "obvious estimates for $\int_0^c e^{-x} dx$" which is not that obvious. Anyway sorry. $\endgroup$ – Najib Idrissi May 30 '11 at 11:47
  • $\begingroup$ Nice! And for free we get $\int_0^c e^{-x}dx$. $\endgroup$ – André Nicolas May 30 '11 at 23:10

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