Vector-valued integration
Kowalski
Motivation. We’ve spent several sections modifying our Calc I interpretations of differentiation concepts
to apply to vector-valued functions. Can we do the same thing for integration?
Vector-valued integration. We’ve already seen that the definite integral of a space curve can be thought
of as a component-wise operation:
"!
#
! b
! b
! b
b
f (t) dt =
x(t) dt,
y(t) dt,
z(t) dt .
f (t) = !x(t), y(t), z(t)"
=⇒
a
a
a
a
Unfortunately, our standard intuitive understanding of the definite integral as the “net signed are under a
curve” makes no sense here, since any reasonable interpretation of “signed area” would be a scalar, whereas
the integral of a vector-valued function is always a vector. Our interpretation of vector-valued functions
as position functions fares no better, since while we have interpretations for the integrals of velocity and
acceleration, we don’t have a physical interpretation of the integral of position.
Two applications. Nevertheless, we can still relate some important aspects of a space curve with vectorvalued integration.
• The displacement vector from f (a) to f (b), that is, the vector starting at the point on the curve
corresponding to t = a and ending at the point corresponding to t = b, can be determined by integrating
the velocity function:
$
% ! b
displacement vector
=
f ! (t) dt = f (b) − f (a).
from t = a to t = b
a
Note that this is just the Fundamental Theorem of Calculus, applied to a vector-valued function.
&b
• Recall from Calc I that if p(t) is a position function of particle moving in a line, then a |p! (t)| dt
represents the total distance traveled by the particle, rather than just the net displacement. Applying
this concept to a general vector-valued function, it follows that the integral of the speed of a vectorvalued function should represent the total distance traveled by the particle, or, equivalently, the total
arclength of the curve over the interval in question:
$
% ! b
' ! '
the arclength of the curve
'f (t)' dt
=
from t = a to t = b
a
! b(
)
*2 )
*2 )
*2
=
x! (t) + y ! (t) + z ! (t) dt.
a
• The position vectors f (a) and f (b) are shown below in orange, together with the displacement vector
in purple and the portion of the curve whose arclength is determined by t = a and t = b in brown:
• It is a sad fact, but it is usually impossible to work out the arclength of a space curve exactly, as
it is almost always impossible to find an antiderivative of the speed formula %f ! (t)% with which to
apply the Fundamental Theorem. As a result, arclengths are typically approximated using “numerical
integration” techniques, such as Riemann sums or series expansions.