What is an integral line?

In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms path integral, curve integral, and curvilinear integral are also used; contour integral is used as well, although that is typically reserved for line integrals in the complex plane.

Which is a line segment?

So, a line segment is a piece or part of a line having two endpoints. Unlike a line, a line segment has a definite length. The length of a line segment can be measured either in metric units such as millimeters, centimeters, or customary units like feet or inches.

When can you use Green’s theorem?

Warning: Green’s theorem only applies to curves that are oriented counterclockwise. If you are integrating clockwise around a curve and wish to apply Green’s theorem, you must flip the sign of your result at some point.

What is meant by line integral and path?

A line integral (sometimes called a path integral) is the integral of some function along a curve. One can integrate a scalar-valued function along a curve, obtaining for example, the mass of a wire from its density. One can also integrate a certain type of vector-valued functions along a curve.

Can a line integral be zero?

You can have an integral around a closed loop being 0 without the field being conservative. You just can’t have it being 0 for all closed loops (so it can be a coincidence for one loop).

What is DS equal to?

The quantity ds/dt is called the derivative of s with respect to t, or the rate of change of s with respect to t. It is possible to think of ds and dt as numbers whose ratio ds/dt is equal to v; ds is called the differential of s, and dt the differential of t.

What is the difference between DR and DS?

So ds just means a general path integral, but dr is specifically through normal space. So if your teacher says “ds as a path through space” then it does mean the same as dr.

What is an example of a line integral?

Let’s take a look at an example of a line integral. Example 1 Evaluate ∫ C xy4ds ∫ C x y 4 d s where C C is the right half of the circle, x2 +y2 =16 x 2 + y 2 = 16 traced out in a counter clockwise direction.

How do you evaluate line integrals?

So, when evaluating line integrals be careful to first note which differential you’ve got so you don’t work the wrong kind of line integral. These two integral often appear together and so we have the following shorthand notation for these cases.

Does the direction of a line integral affect its value?

However, there are other kinds of line integrals in which this won’t be the case. We will see more examples of this in the next couple of sections so don’t get it into your head that changing the direction will never change the value of the line integral.

What are some interesting facts about line integrals with respect to arc?

We then have the following fact about line integrals with respect to arc length. So, for a line integral with respect to arc length we can change the direction of the curve and not change the value of the integral. This is a useful fact to remember as some line integrals will be easier in one direction than the other.