The Constant Is a Vector

You'll learnto antidifferentiate a vector-valued function one component at a time, to carry a constant of integration with one entry per slot rather than a single shared scalar, to read an initial condition as two equations, to check an answer by substituting the starting parameter back into both slots, and to tell a displacement from a position and from a distance traveled.

Antidifferentiating a vector is componentwise, exactly like differentiating one, so almost nothing here is new. The single idea that costs points is what comes after: the constant of integration is a vector, with one entry per component and one equation to determine each. The picture is unhelpful on purpose, because the constant slides a curve without changing its shape. Twelve short steps; press Next for each one.

12 STEPS · 6 QUICK CHECKS · ONE CONSTANT PER SLOT, AND THE CHECK THAT CATCHES IT · v2

antidifferentiate each slot, then put the starting parameter back in

Before you start

What you're looking at

One plane at equal scale. A particle has velocity v(t) = ⟨2t, 3t²⟩ for t from 0 to 2, and every curve you see is drawn by integrating that velocity rather than by plotting a formula.

The question

Antidifferentiating gives ⟨t², t³⟩ plus a constant. What kind of object is that constant, and why can the shape of the curve never tell you?

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