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, 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, and either way 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 one, 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. Twelve short steps; press Next for each one.

12 STEPS · 6 QUICK CHECKS · BC ONLY · A TOTAL THAT SETTLES, OR ONE THAT DOES NOT · v2

The edge slides, the total piles up, and only some totals arrive

Before you start

What you're looking at

Two panels on one horizontal scale. The upper one is a curve with the area under it shaded out to an edge that moves; the lower one is the total collected so far.

The question

As that edge runs out to infinity, or up to a point where the curve has no top, does the total settle on a number?

Step 1 / 12