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 June 12th, 2014, 04:25 PM #1 Newbie   Joined: Jun 2014 From: Buffalo, NY Posts: 2 Thanks: 0 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. June 12th, 2014, 04:54 PM   #2
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Quote:
 Originally Posted by porcupine73 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 $x \to \infty$. June 14th, 2014, 04:36 AM #3 Newbie   Joined: Jun 2014 From: Buffalo, NY Posts: 2 Thanks: 0 Math Focus: infinity 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! Tags equal, finite, infinite, infinity, integrating, number, time Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post Shen Elementary Math 2 June 5th, 2014 07:50 AM thebroker Number Theory 8 October 19th, 2011 05:29 AM Giddyotwiggy Applied Math 1 June 10th, 2011 02:48 AM Agno Number Theory 21 January 31st, 2011 10:04 AM mathdigger Number Theory 5 September 3rd, 2010 09:37 PM

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