Can you explain line integrals

Curve integral

In this article we mainly want to show you what a Curve integral is and how you can calculate it. As a basis for this, we will first explain the termspath andCurve. In addition, we explain how to get the formula for calculating curve integrals and discuss important properties of curve integrals. Two calculation examples serve to illustrate the theoretical considerations.

Ours is even clearer and more memorable Video on the subject Curve integral.

Curve integral simply explained

From school you are sure to have the ordinary integral over a function known.

The Definition set the function is a subset of the real numbers. To calculate the integral, imagine that you all Function values between the borders and sum up.

However, you can also integrate functions that have a Subset of as a definition set have.

However, it is not possible to simply specify two integral limits here. Instead, a subset of can be specified as the integration area. If you have a so-called Curve selects, then the considered integral is called Curve integral.

For example, the domain can be the whole his and the looked at Curve a circle be in the plane. Then you can imagine that for the calculation of the curve integral all Function valuesalong the circlesummed up Need to become.

Path and curve

In order to be able to understand exactly what a curve integral is, we want the term Curve explain which is closely related to the concept of Way.

path

A path is a steady mapping of a real interval in the With :

This is called picture the track of the way and the mapping rule is called Parameterization of the way. The following two paths have the same trace.

It represents the unit semicircle in the upper half plane.

Curve

The A trace of a path is called a curve. So put and represents the parameterization of the same curve.

Curve integral 1st type

Curve integrals 1st type are curve integrals of a scalar function. Such a feature will also Scalar field called. She assigns each value a real number to.

Is the subset open and the parameterization of a piecewise continuously differentiable curve. Then is called

the Curve integral 1st type of along the curve . The term des is often used for a curve integral Line integral used. Also the term Path integral is common for this. However, different paths may just as well different parameterizations one and same curve describe, as has already been set out above.

Clear interpretation or derivation

The interpretation of integrals as totalover infinitely fine rectangles be known. The function value represents the height of such a rectangle at the point under consideration. The width of the rectangle corresponds to a small section of the area over which it is integrated. With the curve or line integral, this area is just a curve. This is done using the variable parameterized that between the limits and running. Now let's try that Length of a small piece of curve to be determined approximately. To do this, we divide the parameterization interval into Pieces a:

To the Length of the curve piece between and to approach, we simply consider the length of the straight line through these two points:

Now this equation can be broken down into Mean theorem of differential calculus also express it as follows:

The following applies here .

If one now forms the Riemann sum over all these small curve pieces and moves on to an infinitely fine decomposition and ) that's how you get that Curve integral 1st type.

Calculate curve integral 1. Art

To calculate a Curve integral type 1 you can remember the following procedure:

  1. The Curveparameterize and in deploy
    With
  2. The scalar arc element determine
  3. In the integral with the Limits and use and calculate

Curve integral example 1. Art

The procedure just described, with which a curve integral can be calculated, will now be illustrated using an example. For this we want the curve or line integral of the function along the circle around the origin with radius to calculate.

Curve integral circle:

  1. Parameterize the circle