Let the Finish Line Move

You'll learnto read an accumulation function off the graph of what it accumulates, to see why its derivative is the integrand itself, and to carry the two adjustments — a chain-rule factor and a sign — that a moving limit which is not simply x always brings with it.

Fix the start of a definite integral and let the finish line move. The integral stops being a number and becomes a function of where that finish line is, and the Fundamental Theorem says what its derivative is in five characters: you get the integrand back. Everything expensive in this topic happens when the finish line is not simply x. Then it travels at its own speed, and that speed multiplies. Miss it and the answer is a correct expression for the wrong question. Twelve short steps; press Next for each one.

12 STEPS · 6 QUICK CHECKS · THE HEIGHT AT THE EDGE, TIMES THE SPEED OF THE EDGE · v2

move the finish line, and the integral becomes a function

Before you start

What you're looking at

Two panels on the same horizontal scale. Above is a function with the area under it shaded from a fixed start to a right edge; below is that collected area plotted against where the edge is.

The question

If the edge moves, how fast does the collected area grow — and what changes when the edge is not simply x?

Step 1 / 12