What does it mean to integrate along a curve?

What does it mean to integrate along a curve?

line integral
Vector calculus. In qualitative terms, a line integral in vector calculus can be thought of as a measure of the total effect of a given tensor field along a given curve. The line integral of f would be the area of the “curtain” created—when the points of the surface that are directly over C are carved out.

What does contour integral represent?

A contour integral is a line integral over a closed curve in the complex plane. A contour integral is usually represented using the symbol. Line integrals are integrals whose integrand is to be evaluated along a curve. Line integrals can be over curves within the complex plane or within the real plane.

Does line integral give area?

A line integral allows for the calculation of the area of a surface in three dimensions. Line integrals have a variety of applications. For example, in electromagnetics, they can be used to calculate the work done on a charged particle traveling along some curve in a force field represented by a vector field.

What is contour integration in complex analysis?

In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane. One use for contour integrals is the evaluation of integrals along the real line that are not readily found by using only real variable methods.

What does the circle on the integral mean?

The circle is there to remind us that the domain of integration, whether it be 1D or 2D or whatever, is closed, in just the same way we could put multiple integral signs to remind us how many dimensions we’re in.

Is volume the integral of area?

So, in this case the volume will be the integral of the cross-sectional area at any x , A(x) . Note as well that, in this case, the cross-sectional area is a circle and we could go farther and get a formula for that as well.