Vector-Valued Integration

The integral of a vector-valued function can be defined in terms of Riemann sums as in Chapter 5. We will define it more simply via componentwise integration (the two definitions are equivalent). In other words,

The integral exists if each of the components x(t), y(t), z(t) is integrable. For example,

Vector-valued integrals obey the same linearity rules as scalar-valued integrals (see Exercise 72).

An antiderivative of r(t) is a vector-valued function R(t) such that R′(t) = r(t). In the single-variable case, two functions f1(x) and f2(x) with the same derivative differ by a constant. Similarly, two vector-valued functions with the same derivative differ by a constant vector (i.e., a vector that does not depend on t). This is proved by applying the scalar result to each component of r(t).

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THEOREM 4

If R1(t) and R2(t) are differentiable and , then

R1(t) = R2(t) + c

for some constant vector c.

The general antiderivative of r(t) is written

∫r(t) dt = R(t) + c

where c = 〈c1, c2, c3〉 is an arbitrary constant vector. For example,

Fundamental Theorem of Calculus for Vector-Valued Functions

If r(t) is continuous on [a, b], and R(t) is an antiderivative of r(t), then

EXAMPLE 8: Finding Position via Vector-Valued Differential Equations

The path of a particle satisfies

Find the particle’s location at t = 4 if r(0) = 〈4, 1〉.

Solution The general solution is obtained by integration:

The initial condition r(0) = 〈4, 1〉 gives us

r(0) = 〈2, 0〉 + c = 〈4, 1〉

Therefore, c = 〈2, 1〉 and (Figure 7)

Particle path

The particle’s position at t = 4 is