A Change Is Not a Position

You'll learnto read every integral in this topic as a change rather than a value, to add the initial condition back, to tell displacement from total distance, and to decide whether a particle is speeding up from the signs of v and a together.

Integrating a velocity does not give you a position. It gives you a change in position, and that change is signed. Both halves of that sentence go missing, and each one costs a different point: the first drops the starting value, and the second turns a trip out and back into no trip at all. Everything in this topic is one of those two, wearing different nouns.

8 STEPS · 6 QUICK CHECKS · AN INTEGRAL IS A CHANGE, AND A SIGNED ONE · v1

∫ v dt = displacement ⇒ s(t₂) = s(t₁) + ∫ v dt ⇒ total distance = ∫ |v| dt

Before you start

What you're looking at

The velocity of a particle moving along a line, and the region between that curve and the axis. Area above the axis is forward travel and area below it is backward travel.

The question

Is the question asking how much something changed, or what it now is? And does it want the net result or every metre actually covered?

Watch for

The instants where the curve crosses the axis. Those are where the particle turns around, and they are the only places the interval ever needs splitting.

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