Integrals over the whole real line

Chapter 12 Integrals over the whole real line

Plenty of applications of integration make sense if we try to integrate over intervals like \([3,\infty)\) or \((-\infty,7]\text{.}\) A couple of examples:

Example 12.1.

If we assume (which is a questionable assumption!) that the current economic situation will continue as it is now, and \(R(t)\) represents the annual revenue in year \(t\) (measured in dollars per year), then

\begin{equation*} \displaystyle\int_{10}^{100} R(t)\ dt \end{equation*}

would represent the total revenue collected in years 10 through 100;

\begin{equation*} \displaystyle\int_{10}^{100000}R(t)\ dt \end{equation*}

would represent the total revenue collected from year 10 to year 100,000. We could ask: What is the revenue collected from year 10 on?. That is, we want to integrate \(R(t)\) on the interval \([10,\infty)\text{.}\)

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Example 12.2.

The density \(D(h)\) of particles in the atmosphere, as a function of the altitude \(h\) above the surface of the earth, decays approximately exponentially: \(d(h)=Ce^{-kh}\text{.}\) If we integrate \(D(h)\) over a closed interval like \([0,10]\text{,}\) we get the total number of particles between altitudes \(h=0\) and \(h=10\text{;}\) if we keep increasing the upper limit of the integral, we pick up more and more particles. If we want to know how many particles there are altogether, we need to keep letting \(h\) get bigger and bigger. The natural symbol to represent this would be

\begin{equation*} \displaystyle\int_{0}^\infty Ce^{-kh}\ dh \end{equation*}

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Now, many times the answer to the question "How much revenue will we take in from now until eternity?" is something we’d expect to be "infinitely much". But, as we’ll see, this is not always the case.

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The situation when integrating out to infinity is similar to the situation with infinite sums. Because there is no already assigned situation with infinite sums. Because there is no already assigned meaning for summing infinitely many things, we defined this as a limit, which in each case needs to be evaluated:

\begin{equation*} \displaystyle\sum_{k=1}^\infty a_k := \lim_{M \to \infty} \displaystyle\sum_{k=1}^M a_k \, . \end{equation*}

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It is the same when one tries to integrate over the whole real line. We define such integrals by integrating over a bigger and bigger piece and taking the limit. In fact the definition is even pickier than that. We only let one of the limits of integration go to zero at a time. Consider first an integral over a half-line \([a,\infty)\text{.}\)

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