Integrals of Vector Functions

Definition

The integral of a vector-valued function $\mathbf{r}(t)$ is a vector whose components are the integrals of the original function’s component functions. It represents the accumulation of vector quantities (like displacement or impulse) over an interval.

Why It Matters

We live in a 3D world where forces act in directions. Scalar calculus is insufficient for physics; vector integration is the language of flight paths, planetary motion, and any system where “where” matters as much as “how much.”

Core Concepts

• Indefinite Integral: $\int \mathbf{r}(t) dt = \left( \int f(t) dt \right)\mathbf{i} + \left( \int g(t) dt \right)\mathbf{j} + \left( \int h(t) dt \right)\mathbf{k} + \mathbf{C}$, where $\mathbf{C}$ is a constant vector of integration.

  • How to read: “The integral of the vector function r of t with respect to t is equal to the integral of f of t with respect to t times i-hat, plus the integral of g of t with respect to t times j-hat, plus the integral of h of t with respect to t times k-hat, plus the constant vector C.”
  • Meaning / when to use: Integrate each component separately; the result is a vector antiderivative with one constant per component.

• Definite Integral: The Fundamental Theorem of calculus applies: $\int_a^b \mathbf{r}(t) dt = \mathbf{R}(b) - \mathbf{R}(a)$, where $\mathbf{R}$ is the antiderivative vector function.

  • How to read: “The integral from a to b of the vector function r of t with respect to t is equal to the vector function R of b minus the vector function R of a.”
  • Meaning: Net displacement (or accumulated vector quantity) equals the antiderivative evaluated at endpoints—same FTC logic as scalars, applied component-wise.

• Component-wise Operation: Integration is performed independently in each dimension ($x, y, z$), allowing complex 3D paths to be solved as a series of 1D problems.

Connected Concepts

• Derivative of Vector Functions
• Integrals of Symmetric Functions
• Graphs of Functions: Properties