Line Integrals

Line Integrals

A function can be summed up along a curve. The little distances ds are summed up with an integral, a line integral.

 

[image]

 

The denotion of a line integral of a scalar function f along a curve C is

 

[image]

 

 

The path C can be written in parametric form:

 

[image]

 

The little distances are evaluated by

 

[image]

 

This is the use of the Pythagorean theorem. The length of the hypotenuse is the sum of the squared length of the other sides in a rectangular triangle.

 

 

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Formula of the line integral

 

 

Another writing of the line integral is

 

[image] 

 

This is the product of a vector r and its derivative.

 

[image]  Example

 

Find the line integral of [image] along the path [image] for [image]

 

The x component of the path is [image]. The derivative is [image].

 

The y component of the path is [image]. The derivative is [image].

 

Put in the values for x and y:

 

[image]

 

 

For [image] is 1, the formula shrinks to the terms inside the bracket.

 

 

Integrate and write the boundaries behind the vertical strike:

 

[image]

 

Put in the boundaries in each term:

 

[image]

 

Evaluate the trig terms. Regard the minus sign rule before a bracket:

 

[image]

 

[image]