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June 12th, 2014, 04:25 PM   #1
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Math Focus: infinity
Does integrating a finite number over infinite time equal infinity?

Hi, I was wondering, if I integrate a finite number such as 3 over an infinite amount of time, would the result be infinity? Or does it simply approach infinity but never reach infinity? Thanks.
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June 12th, 2014, 04:54 PM   #2
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Originally Posted by porcupine73 View Post
Or does it simply approach infinity but never reach infinity? Thanks.
This sentence doesn't make sense. Infinity isn't a number, it's not something you can reach.

An integral $\int_a^\infty f(t) dt$ is defined as
$$\lim_{x \to \infty} \int_a^x f(t) dt$$

If this limit is not unique and finite, we say that the integral diverges. We don't say that it approaches infinity or is infinity.

One reason that an integral might diverge is that it's value grows without bound as $x$ grows without bound.

Finally, we do occasionally mention infinity, but the following is no more than a synonym for the above statements. We can say that the integral tends to infinity as $x$ tends to infinty ($x \to \infty$).

In the case of your example
\begin{align*}
\lim_{x \to \infty} \int_a^x c dt &= \lim_{x \to \infty} ct|_a^x \\
&= \lim_{x \to \infty} c(x - a) \\
\end{align*}
And this limit does not exist, the expression $c(x-a)$ diverges. $c(x - a)$ grows without bound as $x \to \infty$. $c(x - a) \to \infty$ as $x \to \infty$.
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June 14th, 2014, 04:36 AM   #3
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Wonderful, thank you, that makes sense, it's been 20 years since I studied that. The limit as x approaches infinity is the wording I was thinking of but how to apply it I couldn't remember. What you presented answered my question, thank you!
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