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9.5 Integrating Vector-Valued Functions

Syllabus
2020
Topic
9.5
Level
—

Integrate Each Rate Component, Then Anchor the Path

If r′(t)=v(t)=⟨vx(t),vy(t)⟩\mathbf r'(t)=\mathbf v(t)=\langle v_x(t),v_y(t)\rangle, then the position vector is found by integrating the horizontal and vertical rates separately. An initial position fixes both constants and selects the one particular solution that describes the motion.

\mathbf r(t)=\mathbf r(t_0)+\int_{t_0}^{t}\mathbf v(u),du=\mathbf r(t_0)+\left\langle\int_{t_0}^{t}v_x(u),du,\int_{t_0}^{t}v_y(u),du\right\rangle

  1. Identify the given rate vector and the time t0t_0 of the initial position.
  2. Integrate each component over the same interval from t0t_0 to tt.
  3. Add the corresponding components of r(t0)\mathbf r(t_0).
  4. Differentiate the result and substitute t=t0t=t_0 to check both conditions.

Example: suppose v(t)=⟨2t,3⟩\mathbf v(t)=\langle2t,3\rangle and r(1)=⟨4,−2⟩\mathbf r(1)=\langle4,-2\rangle. Then
r(t)=⟨4,−2⟩+⟨∫1t2u du,∫1t3 du⟩=⟨4,−2⟩+⟨t2−1,3t−3⟩=⟨t2+3,3t−5⟩.\mathbf r(t)=\langle4,-2\rangle+\left\langle\int_1^t2u\,du,\int_1^t3\,du\right\rangle=\langle4,-2\rangle+\langle t^2-1,3t-3\rangle=\langle t^2+3,3t-5\rangle.
Indeed, r′(t)=⟨2t,3⟩\mathbf r'(t)=\langle2t,3\rangle and r(1)=⟨4,−2⟩\mathbf r(1)=\langle4,-2\rangle, so both the rate vector and initial condition are satisfied.

Do not integrate the speed ∥v(t)∥\|\mathbf v(t)\| when the question asks for position: speed is scalar and its integral gives distance traveled. Integrating the velocity vector gives vector displacement. With indefinite integrals, remember that the two components can have different constants before the initial position determines them.