This table gives a few (3x, y) pairs of a
line in the coordinate plane
56
66
12
58
-28
50
What is the y-intercept of the line?
Answers
Answer:
Learning Objectives
6.2.1 Calculate a scalar line integral along a curve.
6.2.2 Calculate a vector line integral along an oriented curve in space.
6.2.3 Use a line integral to compute the work done in moving an object along a curve in a vector field.
6.2.4 Describe the flux and circulation of a vector field.
We are familiar with single-variable integrals of the form ∫
b
a
f(x)dx, where the domain of integration is an interval [a,b]. 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) and (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.
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.
Scalar Line Integrals
A line integral gives us the ability to integrate multivariable functions and vector fields over arbitrary curves in a plane or in space. There are two types of line integrals: scalar line integrals and vector line integrals. Scalar line integrals are integrals of a scalar function over a curve in a plane or in space. Vector line integrals are integrals of a vector field over a curve in a plane or in space. Let’s look at scalar line integrals first.
A scalar line integral is defined just as a single-variable integral is defined, except that for a scalar line integral, the integrand is a function of more than one variable and the domain of integration is a curve in a plane or in space, as opposed to a curve on the x-axis.
For a scalar line integral, we let C be a smooth curve in a plane or in space and let f be a function with a domain that includes C. We chop the curve into small pieces. For each piece, we choose point P in that piece and evaluate f at P. (We can do this because all the points in the curve are in the domain of f.) We