What is line integral with respect to x?

The line integral of f with respect to x is, ∫Cf(x,y)dx=∫baf(x(t),y(t))x′(t)dt. The line integral of f with respect to y is, ∫Cf(x,y)dy=∫baf(x(t),y(t))y′(t)dt. Note that the only notational difference between these two and the line integral with respect to arc length (from the previous section) is the differential.

What is the formula for line integral?

Line Integral Formula Here, r: [a, b]→C is an arbitrary bijective parametrization of the curve. r (a) and r(b) gives the endpoints of C and a < b. F[r(t)] . r'(t)dt.

What is line integral example?

Let’s take a look at an example of a line integral. Example 1 Evaluate ∫Cxy4ds ∫ C x y 4 d s where C is the right half of the circle,x2+y2=16 x 2 + y 2 = 16 traced out in a counter clockwise direction. We first need a parameterization of the circle.

What does the line integral represent?

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 does a line integral represent?

What is the line integral of a function?

The line integral is then, Don’t forget to plug the parametric equations into the function as well. If we use the vector form of the parameterization we can simplify the notation up somewhat by noticing that, where ∥∥→r ′(t)∥∥ ‖ r → ′ ( t) ‖ is the magnitude or norm of →r ′(t) r → ′ ( t). Using this notation, the line integral becomes,

What happens to the line integral when you change the direction?

So, it looks like when we switch the direction of the curve the line integral (with respect to arc length) will not change. This will always be true for these kinds of line integrals. However, there are other kinds of line integrals in which this won’t be the case.

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.

What is the line integral of F with respect to arc length?

Because of the ds d s this is sometimes called the line integral of f f with respect to arc length. We’ve seen the notation ds d s before. If you recall from Calculus II when we looked at the arc length of a curve given by parametric equations we found it to be, It is no coincidence that we use ds d s for both of these problems.