Line Integrals
Line integral of a function
Line integral of a function in 2D space
Suppose is a function defined on a domain in and is a smooth curve in the domain with arc length parametrization , where ( is the arc length of ). The curve is divided into parts such that is the arc length of the subdivision and is the The line integral of over is
The line integral is the surface area of the "sheet" under the graph of over the smooth curve , as shown in the applet below.
Suppose we are given a parametrization of the smooth curve for , which is not necessary an arc length parametrization. We already know that for the arc length function of the curve , i.e. . Therefore, we have
where .
Example: Find the surface area of the cylinder between xy plane and the parabolic cylinder .
Answer:
The required surface area is the surface area of the "sheet" under the parabolic cylinder over the circle . We can parametrize the circle by for . Then
Therefore, the required surface area can be computed as follows:
The applet below is the illustration of the surface.
Exercise: Let . Evaluate , where is the right half of the circle centered at the origin with radius traced out in anti-clockwise direction.
Line integral of a function in 3D space
Suppose is a function defined on a domain in containing a smooth curve , for . Then the line integral of over is
where .
Example: Evaluate , where is the line segment from to .
Answer:
We first need to parametrize the line segment : Let , for . Then . Therefore, we have
.
Question: Will the value of the line integral change if the line integral is over the line segment from to ?
Exercise: Evaluate , where is the helix given by for .
Line integral of a vector field
Let be a vector field in defined on a domain containing a smooth oriented curve parametrized by arc length . Let be the unit tangent vector at each point of . The line integral of is
Consider any given parametrization of (not necessary an arc length parametrization) for . Since , we have the following
Other notations for the above line integral:
Note: In some textbooks, is called the line integral of with respect to and
Similarly, is called the line integral of with respect to and
and hence .
Example: Evaluate , where is the line segment from to .
Answer:
First, we parametrize : for . Then .
.
We can define the line integral of a 3D vector field over a 3D smooth oriented curve in a similar way: Suppose be a 3D vector field and is a smooth oriented curve parametrized by for . Then the [b]line integral of over is
Exercise:
Properties of line integrals
Reversing the orientation of a curve
Let be a smooth curve in 2D (or 3D) space with a parametrization for , which determines the orientation of the curve (the direction that is traced out as increases). We define to be the curve with the same points as , but the orientation is reversed. In fact, we can parametrize as follows:
for .
As you can see, as increases from to , the curve is traced from to i.e. in reverse direction.
For any function , its line integral over is
(use the substitution )
Therefore, reversing the direction of the curve does not change the value of the line integral of a function i.e. .
Now we consider any 2D vector field . its line integral over is
(use the substitution )
Therefore, we have .
Line integrals over piecewise smooth curves
We can define a line integral of a vector field (or function) over a curve that is piecewise smooth i.e. is a curve formed by connecting a finite number of smooth curves together. Suppose is any 2D vector field. Then we have
Example: Evaluate , where is the loop along the triangle from to to and back to .
Answer:
Consider the following parametrization of :
for ( to )
for ( to )
for ( to )
Then we have