yifli said:There are line integral with respect to arc length and line integral with respect x/y.
I know [itex]\int_C Pdx+Qdy[/itex] is useful to calculate the work. When do we need the line integral with respect to arc length?
A line integral with respect to arc length is a type of integration that calculates the length of a curve in a given direction. It is used to find the distance traveled along a curve by taking into account the shape and curvature of the curve.
The main difference between line integrals with respect to arc length and regular integrals is that the former takes into account the shape and direction of the curve, while the latter only considers the two-dimensional area under the curve. Line integrals with respect to arc length also require a different set of mathematical tools and techniques compared to regular integration.
Line integrals with respect to arc length are commonly used in physics and engineering to calculate work done by a force along a curved path. They are also used in calculus to find the length of a curve, as well as in other mathematical applications such as finding the center of mass of a curved object.
Yes, a line integral with respect to arc length can be negative if the direction of integration is opposite to the direction in which the curve is traversed. This can happen if the curve changes direction multiple times or if the curve is traced in a clockwise direction instead of a counterclockwise direction.
Line integrals with respect to arc length can only be used for smooth and continuous curves. They cannot be applied to curves that have sharp corners or discontinuities. Additionally, the calculation of line integrals with respect to arc length can be complex and time-consuming, making it difficult to find an exact solution for more complicated curves.