Calc 3 Lecture NotesSection 14.2Page 1 of 8

Section 14.2: Line Integrals

Big idea: Line integrals are used to compute how a given quantity “adds up” along a curved path.

Big skill: You should be able to compute line integrals (using the evaluation theorem).

Introductory examples:

  1. Compute the mass of a baseball bat with linear density for 0 x 1 m. (Note that in this example, we are adding up a quantity over a curve, which is simply a straight segment of length 1 m.)
  1. Compute the mass of a helix of radius 1 m and that has 5 turns along 1 m of its axis, and a linear mass density that increases linearly from 1 g/m to 26 g/m over its length. Notice that the total arc length of this helix is: , so we can write the linear mass density as a function of arc lengths as:fors over the interval
    0 s 31.432 m.

Note that in this example, we had to make an effort to relate the density to the arc length. This is not usually how things are done. Instead, we parameterize the curve and leave the integrand alone…
Definition 2.1: Line Integral with Respect to Arc Length

The line integral of f(x, y, z) with respect to arc length along the oriented curve C in three-dimensional space, written as is defined by

,

provided the limit exists and is the same for all choices of evaluation points.

Theorem 2.1: Evaluation Theorem (for Line Integrals)

If f(x, y, z) is continuous in a region D containing a curve C, C can be described parametrically by (x(t), y(t), z(t)) for atb, and x(t), y(t), and z(t) have continuous first derivatives, then

.

Likewise, if f(x, y) is continuous in a region D containing a curve C, C can be described parametrically by (x(t), y(t)) for atb, and x(t) and y(t) have continuous first derivatives, then

.

Proof:
Practice:

  1. Compute the mass of the helix from page 1 given that . Note that you get the same answer.
  1. Evaluate , given a curve C specified by x = cos(t),y = sin(t), and z = cos(t) for 0 t 2.
  1. Evaluate , given a curve C specified by the quarter ellipse in the first quadrant of the x-y plane with a semi-major axis of length 2 aligned along the x-axis and semi-minor axis of 1.

A curve C is called smooth if it can be described parametrically by (x(t), y(t), z(t)) for atb, x(t), y(t), and z(t) have continuous first derivatives, and on the interval [a, b].

For example, the curve C = (t2, t3, t2) is not smooth on the interval [-1, 1] because when t = 0:

This means that the line integral is undefined, but if we break up the curve into two subpieces C = C1C2 over the intervals [-1,0] and [0, 1], then we can break up the integral into two subpieces as well. A curve that can be subdivided into smooth subpieces is called piecewise smooth.

Theorem 2.2: Line Integrals on a Piecewise Smooth Curve

If f(x, y, z) is continuous in a region D containing an oriented curve C, and Cis piecewise-smooth with C = C1C2 … Cn and all the Ci are smooth and the terminal point of Ci is the initial point of Ci+1 for all i = 1, 2, … n-1, THEN

AND

Practice:

  1. Evaluate , given a curve C specified by a sector of outer radius 2 and inner radius 1 and angular extent of /2 symmetric to the x-axis.

Geometric interpretation of the line integral:

Just as is the area bounded by x = a, x = b, y = 0 and y = f(x), is the area of the “vertical” surface bounded by the “vertical” line through (x(a), y(a)), the vertical line through (x(b), y(b)), the parametric curve (x(t), y(t), 0), and the parametric curve (x(t), y(t), f(x(t), y(t)) ).

Theorem 2.3: Arc Length of a Curve from a Line Integral

For any piecewise-smooth curve C, gives the arc length of the curve C.

Next, we worry about evaluating a line integral where the integrand is the dot product of a vector field and the differential tangent vector to the curve, as arises in the computation of work:

Definition 2.2: Component-Wise Line Integrals

The line integral of f(x, y, z) with respect to xalong the oriented curve C in three-dimensional space is written as and is defined by:

,

provided the limit exists and is the same for all choices of evaluation points.

The line integral of f(x, y, z) with respect to y along the oriented curve C in three-dimensional space is written as and is defined by:

,

provided the limit exists and is the same for all choices of evaluation points.

The line integral of f(x, y, z) with respect to z along the oriented curve C in three-dimensional space is written as and is defined by:

,

provided the limit exists and is the same for all choices of evaluation points.

Theorem 2.4: Evaluation Theorem (for Component-Wise Line Integrals)

If f(x, y, z) is continuous in a region D containing a curve C, C can be described parametrically by (x(t), y(t), z(t)) for atb, and x(t), y(t), and z(t) have continuous first derivatives, then

.

Theorem 2.5: Evaluation Theorem (for Component-Wise Line Integrals)

If f(x, y, z) is continuous in a region D containing an oriented curve C, then:

If C is piecewise-smooth:

If C is piecewise-smooth and forms a closed loop:

Consequence: