Mathematical Structure
Integrating Vector-Valued Functions
Prerequisites:
The definite integral of a vector-valued function
is defined in exactly the same way as the derivative of an ordinary function.
where
A proof similar to the proof in the module on
Differentiating
Vector-Valued Functions shows that for a vector-valued function
whose values are in R^n the definite integral may be found by
integrating each of the coordinates -- that is,
The same thing is true for the indefinite integral, or the antiderivative,
The CAS files below show how each computer algbera system can integrate
vector-valued functions.
Check Your Understanding
Find the indefinite integral of each of the following functions.
- F(t) = (sin t, cos t, 3 t)
- F(t) = (2 t + 2, 3 t + 4, 4 t + 5)
- F(t) = (1 - t^2, 12 t^3, 3 t)
Copyright c 1995 by
Frank Wattenberg, Department of Mathematics, Montana State University,
Bozeman, MT 59717