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9.5 Integrating Vector-Valued Functions

Part of Parametric Equations, Polar Coordinates, and Vector-Valued Functions in AP® Calculus BC. Read the opening section, then create a free account for the full interactive lesson, practice, and progress tracking.

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To integrate a vector-valued function, you integrate each component separately. If
r′(t)=⟨f(t),g(t)⟩\mathbf{r}'(t)=\langle f(t),g(t)\rangle
then
r(t)=⟨∫f(t) dt,∫g(t) dt⟩\mathbf{r}(t)=\langle \int f(t)\,dt,\int g(t)\,dt\rangle
plus constants in each component. In three dimensions, the same idea applies to all three coordinates. This process reverses differentiation component by component.

You use vector integration most often to recover position from velocity or velocity from acceleration. Since each coordinate acts like its own ordinary function, the main challenge is remembering that constants of integration belong in each component. If an initial condition is given, you use it to solve for those constants just as you would in one-variable calculus.

In this unit

9 lessons

  1. 9.1Defining and Differentiating Parametric EquationsPreview lesson
  2. 9.2Second Derivatives of Parametric EquationsPreview lesson
  3. 9.3Finding Arc Lengths of Curves Given by Parametric EquationsPreview lesson
  4. 9.4Defining and Differentiating Vector-Valued FunctionsPreview lesson
  5. 9.5Integrating Vector-Valued FunctionsThis lesson
  6. 9.6Solving Motion Problems Using Parametric and Vector-Valued FunctionsPreview lesson
  7. 9.7Defining Polar Coordinates and Differentiating in Polar FormPreview lesson
  8. 9.8Find the Area of a Polar Region or the Area Bounded by a Single Polar CurvePreview lesson
  9. 9.9Finding the Area of the Region Bounded by Two Polar CurvesPreview lesson

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