## Defenite Integral. Change of Variables.

Limits, differentiation, related rates, integration, trig integrals, etc.
sepoto
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### Defenite Integral. Change of Variables.

$\lim_{x \to +\infty}E(x)=\frac{1}{\pi}*\frac{\sqrt{\pi}}{2}=\frac{1}{2}$

The calculation for the limit as x approaches infinity is above and it is supposed to be 1/2. I'm trying to figure out how that limit was calculated. The solution for the limit is not really enumerated enough that I can figure out how 1/2 was arrived at.

Thank You!

stapel_eliz
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They show all the steps. At which point does the worked solution stop making sense?

sepoto
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### Re: Defenite Integral. Change of Variables.

The limit as x approaches infinity of the definite integral below (I hope I have the right definite integral, there are two others to choose from in this problem):

$\frac{1}{\sqrt{\pi}}\int^{x/\sqrt{2}}_{0}e^{-t^{2}}dt$

My solutions manual shows the result of the calculation to be:
$\lim(x \to \infty)=\frac{1}{\pi}*\frac{\sqrt{\pi}}{2}=\frac{1}{2}$

The part I don't understand right now is how this was calculated:
$\frac{1}{\pi}*\frac{\sqrt{\pi}}{2}$

I don't see how $1/\pi$ was reached. I don't see how $\frac{\sqrt{\pi}}{2}$ was arrived at either. Could someone fill me in on how that limit was calculated?

Thank You!

stapel_eliz
Posts: 1628
Joined: Mon Dec 08, 2008 4:22 pm
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The exercise statement gives you the following:

. . . . .$\displaystyle{\lim_{x \rightarrow \infty}\, \int_0^x\, e^{-t^2}\, dt\, =\, \frac{\sqrt{\pi}}{2}}$

So there is no "computation" or "figuring" to do. This is just something you are given. You are also given the integral expression for E(x) as being something that occurs in probability and statistics. For the origin of this expression, you'd likely need to study probability or statistics.

In other words, there's nothing to figure out or compute or develop or explain; you're just using what they've given you.

Note: There appears to be a typo in the solution, as the $\frac{1}{\sqrt{\pi}}$ in the original expression suddenly changes to $\frac{1}{\pi}$ at the end of the worked solution to part (a), which of course would prevent the simplification to $\frac{1}{2}$. Draw in the square-root symbol, and the proof should be fine.