Integrating Vector-Valued Functions

Definition

Integrating a vector-valued function involves finding the antiderivative (or integral) of each component of the vector separately. It calculates the area under the curve represented by the vector.

Analogy

Imagine pouring water into different cups placed along a table. Each cup represents one component of the vector, and integrating the vector-valued function is like measuring how much water accumulates in each cup over time.

Related terms

Arc Length Parameterization: An arc length parameterization describes a curve using its arc length as an independent variable.

Line Integral: A line integral calculates quantities such as work or circulation along a curve defined by a vector field.

Fundamental Theorem for Line Integrals: This theorem relates line integrals to antiderivatives and provides shortcuts for evaluating certain types of line integrals.