Evaluate the line integral where c is the given curve

Q: Evaluate the line integral, where C is the given curve.

In the previous two sections we looked at line integrals of functions. In this section we are going to evaluate line integrals of vector fields. Note the notation in the integral on the left side. That really is a dot product of the vector field and the differential really is a vector. We can also write line integrals of vector fields as a line integral with respect to arc length as follows,. If we use our knowledge on how to compute line integrals with respect to arc length we can see that this second form is equivalent to the first form given above. In general, we use the first form to compute these line integral as it is usually much easier to use.

Evaluate the line integral where c is the given curve

We have so far integrated "over'' intervals, areas, and volumes with single, double, and triple integrals. We now investigate integration over or "along'' a curve—"line integrals'' are really "curve integrals''. As with other integrals, a geometric example may be easiest to understand. What is the area of the surface thus formed? We already know one way to compute surface area, but here we take a different approach that is more useful for the problems to come. As usual, we start by thinking about how to approximate the area. We pick some points along the part of the parabola we're interested in, and connect adjacent points by straight lines; when the points are close together, the length of each line segment will be close to the length along the parabola. If we add up the areas of these rectangles, we get an approximation to the desired area, and in the limit this sum turns into an integral. Then as we have seen in section Example Now we turn to a perhaps more interesting example.

Approximation 5.

Evaluate the line integral, where C is the given curve. Use a calculator or CAS to evaluate the line integral correct to four decimal places. Short Answer Step-by-step Solution. Now share some education! Short Answer Expert verified. Step by step solution

Such a task requires a new kind of integral, called a line integral. Line integrals have many applications to engineering and physics. They also allow us to make several useful generalizations of the Fundamental Theorem of Calculus. And, they are closely connected to the properties of vector fields, as we shall see. A line integral gives us the ability to integrate multivariable functions and vector fields over arbitrary curves in a plane or in space.

Evaluate the line integral where c is the given curve

Such an interval can be thought of as a curve in the xy -plane, since the interval defines a line segment with endpoints a , 0 a , 0 and b , 0 b , 0 —in other words, a line segment located on the x -axis. Suppose we want to integrate over any curve in the plane, not just over a line segment on the x -axis. Such a task requires a new kind of integral, called a line integral.

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That is, reversing the orientation of a curve changes the sign of a line integral. It is completely possible that there is another path between these two points that will give a different value for the line integral. Collapse menu Introduction 1 Analytic Geometry 1. Recall that in the simplest case, the work done by a force on an object is equal to the magnitude of the force times the distance the object moves; this assumes that the force is constant and in the direction of motion. The result is the scalar line integral of f f along C. Find the work done. Note that in a scalar line integral, the integration is done with respect to arc length s , which can make a scalar line integral difficult to calculate. A: Line integral. Non-necessary Non-necessary. Advanced Engineering Mathematics. Triple Integrals 6. McGraw-Hill Education.

In this section we are now going to introduce a new kind of integral. However, before we do that it is important to note that you will need to remember how to parameterize equations, or put another way, you will need to be able to write down a set of parametric equations for a given curve.

Calculate the flux across C. In fact, we will be using the two-dimensional version of this in this section. To see where the term circulation comes from and what it measures, let v represent the velocity field of a fluid and let C be an oriented closed curve. You also have the option to opt-out of these cookies. Comparison Tests 6. Functions of Several Variables 2. Is the work the same along the two paths? Out of these, the cookies that are categorized as necessary are stored on your browser as they are essential for the working of basic functionalities of the website. Nathan Klingbeil. Hyperbolic Functions 5 Curve Sketching 1. A piecewise smooth curve C consists of a finite number of smooth curves that are joined together end to end. Calculus Volume 3 6. Such a task requires a new kind of integral, called a line integral.