Section 18.1 The Idea of a Line Integral

If you can imagine having a boat in our gulf stream example from last chapter, depending on which direction you are heading you may be with the current, against it, or at some angle with or against it A line integral determines to what extent a curve in a vector field is going with the vector field or against it A curve is said to be oriented if its direction is specified

Suppose we have a vector field,, and an oriented curve, C in the vector field –We will approximate our curve with little (straight) line segments along which is approximately constant –We will find the value of on each line segment –Thus for each line segment we get the dot product –Summing over all the pieces gives us the Riemann sum –Taking the limit gives us

A line integral of a vector field along C is If the vector field is a force field then our line integral gives us the amount of work done by the field on the object as it moves along C Since the line integral uses a dot product –It is positive if they are in the same direction –It is negative if they are in opposite directions –It is zero if they are perpendicular

Alternative Notation for Line Integrals In 2 space Similarly, in 3 space we have so

Examples Calculate the following line integrals