SUNY Geneseo Department of Mathematics

Scalar Line Integral Examples

Wednesday, April 19

Math 223
Spring 2023
Prof. Doug Baldwin

Anything You Want to Talk About?

Problem set 11, problem 2 (show that a double integral of a product of functions is equal to the product of 2 single-variable integrals)?

The heart of the proof is to write the original integral as an iterated integral and then look for constants that can be factored out of parts of it:

Double integral of F of X times G of Y can factor G out of integral of F then that integral out of the integral of G

Examples of Scalar Line Integrals

3 Variables

Find the integral of w = 2√y (x² + z²) along the curve r(t) = ⟨ sin t, t², cos t ⟩from t = 0 to t = 1.

The first step is to write the integral in the calculation form:

Integral along R of W with respect to S is integral from T 1 to T 2 of F of R of T times magnitude of R prime

To evaluate this, you need to know r′ and its magnitude:

R prime is vector cosine T, 2 T, negative sine T; magnitude is square root 1 plus 4 T squared

Now you can plug things into the calculation form of the integral and evaluate. Notice that the “f(r(t))” part means to write out f, but with x, y, and z replaced with their corresponding definitions from r, i.e., the x, y, and z components of that vector:

Evaluating scalar line integral

Non-Smooth Paths

Can you even integrate g(x, y) = xy² along the checkmark-shaped path from (-1, 0) to (0, 0) and then to (1, 2) at all?

Function G of X and Y is X times Y squared; checkmark-shaped path

You can, thanks to a theorem that says you can break the curves for line integrals into sections, each touching the next only at their ends, and together covering the whole original curve. The original line integral is then the sums of the integrals along the sections:

Integral along R of G equals integral along R 1 of G plus integral along R 2 of G

If so, what’s the integral?

Start by figuring out parametric forms for the two parts of the curve. While you’re at it, you might as well figure out the magnitudes of the derivatives of the parametric curves, since you’ll need that later:

R 1 is vector T, negative T; magnitude of R 1 prime is root 2; R 2 is vector T, 2 T with derivative 1, 2.

Finally, plug the parametric forms into the function, multiply by the magnitudes of the derivatives, and evaluate:

Evaluating line integral as sum of 2 integrals over paths disjoint except at origin

Line integrals really pay off when the function you’re integrating is vector valued. So we should talk about multivariable vector-valued functions, aka “vector fields.”

Please read “Examples of Vector Fields,” “Vector Fields in R²,” “Drawing a Vector Field,” and “Vector Fields in R³” in section 5.1 of the textbook.

We’ll also look at plotting vector fields with Mathematica, so bring it if you want to try out the things we talk about.

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