In
Preview Activity 12.5.1, you approximated the distance traveled for various paths and multiplied by the density of the copium on each piece of the path. In contrast to the line integral of a vector field, the calculations of the ore mined does not depend on what direction the path was traveled. We will now use these same ideas to give precise meaning to the measurement of the acculumation of a scalar function’s output over a path in space.
Let
be a contiunuous function of
and
for some open set around
a curve from a point
to a point
We will begin to approximate the acculumlation of the output of
over
by breaking
into pieces with boundary points
The curve
is the part of
that goes from
to
and
is the displacement vector from
to
We can approximate the accumulation of over with the following sum
where is the output of for some As this sum uses more pieces and all of the lengths of the pieces goes to zero (i.e. ), we would expect that the sum will approach the actual acculumation of over Notice that it won’t matter how we select the point that is used in each piece to evaluate the output of since evaluating the limit as the length of gets smaller will ensure that the output value choosen will be within a shrinking error from the average value on each piece. Evaluating the limit of the sum above as the size of all of the pieces goes to zero will transform our Riemann sum into an integral that will measure the accumulation of the output of over
The notation for a scalar line integral (
) may not immediately make sense. As with the other types of integration we have done (single variable integration, double integrals, line integrals of vector fields, etc.), the subscript of the integral symbol denotes the region of integration. In the case of a scalar line integral, the region of integration is a collection of points given by a curve in space. The function we are integrating is
a scalar-valued functiton of multiple variables. The differential
may seem unusual to you. If you remember from
Section 9.8,
is the arc length of a curve in space. So the differential
in the scalar line integral notation means that we are adding up the output of
over steps in arc length. This should make sense in terms of how we set up our Riemann sums. We did not set up the pieces of our curve as steps in
or
but rather as steps in arc length (estimated by
). This may feel similar to situations such as double integrals, where we generically used
for the differential, but the different contexts called for different values of
For instance, in polar coordinates, we use