Write the Limit, Then Evaluate

You'll learnto spot both ways an integral becomes improper, to write the limit before anything is substituted into anything, to decide convergence by watching a running total rather than by watching the integrand, and to find the blow-up that sits strictly inside an interval and announces itself nowhere in the notation.

An integral is improper when the interval runs to infinity or the integrand blows up somewhere on it. Either way the definition of a definite integral stops applying and a limit takes over. What makes this the quietest error in the unit is that skipping the limit still produces a number — an ordinary-looking, plausible number, arrived at by correct arithmetic. Two pictures settle every question here: an area collected out to a moving edge, and the running total of that area plotted against where the edge has reached. One of those totals bends onto a line. The other never arrives anywhere.

8 STEPS · 6 QUICK CHECKS · BC ONLY · A TOTAL THAT SETTLES, OR ONE THAT DOES NOT · v1

∫ from 1 to ∞ means the LIMIT of ∫ from 1 to b · watch the running total

Before you start

What you're looking at

Two panels on the same horizontal scale. The upper one is an integrand, with the area under it shaded from a fixed start out to an edge that moves. The lower one plots the total collected so far against where that edge has reached, so the height of the lower curve at any b is the amount of shading standing to its left.

The question

As the edge runs out to infinity, or up to a point where the integrand is unbounded, does that running total settle on a number?

Watch for

Blue is the member whose total settles; pink is the member whose total runs away. A dashed yellow line marks what a settling total is heading for. The edge is never at infinity and never at the bad point — that is what the limit is for.

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