Top minus bottom

You'll learnto turn a definite integral into arithmetic with the Fundamental Theorem, Part 2 — why any antiderivative you like will do, why the subtraction is where nearly every mark is lost, and what to do when the integrand changes formula partway across the interval.

Part 1 said that differentiating an accumulation hands the integrand back. Part 2 runs the other way and turns every definite integral into arithmetic: find an antiderivative, substitute both limits, subtract. The theorem itself is not where marks are lost. The subtraction is — and almost always when the value at the lower limit sits below the axis, where a dropped minus sign turns an addition into a subtraction and nothing on the page looks wrong.

8 STEPS · 6 QUICK CHECKS · FIND F, SUBSTITUTE, SUBTRACT — IN THAT ORDER · v1

the integral from a to b of f = F(b) − F(a) ⇒ any antiderivative ⇒ the constant cancels

Before you start

What you're looking at

The two panels from Topic 6.4, on one horizontal scale. The upper one is the integrand and the shaded region under it is signed area. The lower one is an antiderivative of it. What is new is that the lower curve is no longer pinned to zero anywhere — any member of the family will do, and Step 2 draws three of them.

The question

The shaded area is a number. Which two heights on the lower curve does it equal the distance between, and in which order do they come?

Watch for

The vertical bar in the lower panel is the whole topic. Its foot sits below the axis, because the antiderivative is negative at the lower limit — which is precisely where the subtraction goes wrong.

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