Improper Integrals II

I’ve got one last thing to wrap up in my coverage of integration. Well, I never talked about Brown v. Board of Ed., but most curricula these days don’t cover the technique “by court order” anymore.

Did I get the title of the post wrong? No. It turns out that I covered half of this topic two years ago as I prepared to push into infinite series. Back then, I dealt with what happened when we wanted to integrate over an infinite interval. Then I defined an infinite series to be what happened when we used a particular integrator in our Riemann-Stieltjes integral. This could also be useful in setting up multiple integrals over $n$ -dimensional intervals that extend infinitely far in some direction.

But there are other places that so-called “improper integrals” come up, like the example I worked through the other day. In this case, the fundamental theorem of calculus runs into trouble at the endpoints of our interval. Indeed, we ask for an antiderivative on a closed interval, but to get the endpoints we need to have the antiderivative defined on some slightly larger open interval, and it just isn’t.

So here’s what we do: let $f$ be defined on $(a,b]$ and integrable with respect to some integrator $\alpha$ over the interval $[x,b]$ for all $x\in(a,b]$ . Then we can define each integral

$\displaystyle\int\limits_x^bf\,d\alpha$

Just as before, we define the improper integral to be the one-sided limit

$\displaystyle\int\limits_{a^+}^bf\,d\alpha=\lim\limits_{x\rightarrow a^+}\int\limits_x^bf\,d\alpha$

If this limit exists, we say that the improper integral converges. Otherwise we say it diverges. Similarly we can define the improper integral

$\displaystyle\int\limits_a^{b^-}f\,d\alpha=\lim\limits_{x\rightarrow b^-}\int\limits_a^xf\,d\alpha$

taking the limit as $x$ approaches $b$ from the left.

Just as for the first kind of improper integral, we have analogues of the direct comparison and limit comparison tests, and of the idea of absolute convergence. Each of these is exactly the same as before, replacing limits as $x$ approaches $\infty$ with limits as $x$ approaches some finite point from the left or right.

We can also combine improper integrals like we did before to integrate over the whole real line. For example, we could define

$\displaystyle\int\limits_{a^+}^{b^-}f\,d\alpha=\lim\limits_{\substack{x\rightarrow a^+\\y\rightarrow b^-}}\int\limits_x^yf\,d\alpha$

$\displaystyle\int\limits_{a^+}^\infty f\,d\alpha=\lim\limits_{\substack{x\rightarrow a^+\\y\rightarrow\infty}}\int\limits_x^yf\,d\alpha$

As before, we must take these limits separately. Indeed, now we can even say more about what can go wrong, because these are examples of multivariable limits. We cannot take the limit as $x$ and $y$ together approach their limiting points along any particular path, but must consider them approaching along all paths.

In practice, it’s pretty clear what needs to be done, and when. If we have trouble evaluating an antiderivative at one endpoint or another, we replace the evaluation with an appropriate limit.

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