We now discuss the concept of a scalar line integral. This is one of several ways in which integrals of functions of one variable can be generalized to functions of several variables. Here, instead of integrating over an interval on the axes, we now integrate over a curve . The integrand is then a function such that .
Riemann integrals of functions are defined as the limit of Riemann sums
where , where for and where . That is to say that we partition the interval and multiply the function value in a subinterval with the length of this subinterval, namely and make the partition finer and finer.
Analogously we define scalar line integrals. Here, the domain is not an interval , but a curve. Again, the idea is to partition the curve into subcurves and compute the sum of some function value , where is a point on the subcurve , multiplied by the length of the subcurve . In other words,
where in the limit the length of the longest subcurve tends to .
The following result shows how scalar line integrals can be computed.
Scalar line integral
Let be a parameterisation of the curve . Assume that is continuously differentiable and let be continuous such that . Then the scalar line integral is given by
The integral does not depend on the parameterisation of the curve.
If for all , then
is the length of the curve .
Let the curve be parameterised by where and let . Calculate .
Let the curve be parameterised by for and let . Calculate by interpreting the integral as an integral along the -axis.
Question Using the geometrical interpretation of integrals as the area under the graph of , how can you geometrically interpret the scalar line integral then?
Let a curve be parameterised by given by . Further let be given by . Calculate the line integral using a geometrical interpretation of scalar line integrals.
Question Let be a continuously differentiable function. Can you find a method to calculate the length of the graph of ?
Some properties of scalar line integrals
Let be curves and be functions defined on these curves. Let . Let denote the curve which consists of both, and .
Scalar line integrals can be used to calculate for instance the mass or center of mass of a wire. Let be a curve in and let denote the density of the wire at a point on the curve.
- Mass: where denotes the density.
- First moments about coordinate planes: and
- Center of mass: and