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.

8 STEPS · 6 QUICK CHECKS · THE HEIGHT AT THE EDGE, TIMES THE SPEED OF THE EDGE · v1

g(x) = ∫ from a to x of f ⇒ g′(x) = f(x) ⇒ a moving edge carries its own speed

Before you start

What you're looking at

Two panels stacked on the same horizontal scale. The upper one is a function; the shaded region under it is area collected from a fixed start to a right edge. The lower one plots that collected area against where the edge is, so the lower curve is built out of the upper region.

The question

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

Watch for

Area under the axis is drawn in a different colour and counts negatively. The vertical rule linking the two panels is only meaningful while the moving limit is x itself; two steps here take it away on purpose.

Step 1 / 8