SUNY Geneseo Department of Mathematics

Scalar Line Integrals, Part 2

Monday, November 14

Math 223 01
Fall 2022
Prof. Doug Baldwin

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Scalar Line Integrals

Continuing the examples from Friday, based on “Scalar Line Integrals” in section 5.2 of the textbook.

Example

Evaluate the integral of f(x,y,z) = 2√y (x2 + z2) over the curve r(t) = ⟨ sin t, t2, cos t ⟩, 0 ≤ t ≤ 1.

The first step is to figure out r′ and its magnitude:

R of T is vector sine T comma T squared comma cosine T. Derivative's magnitude is radical 4 T squared plus 1

Then plug f(r(t)) and |r′(t)| into the scalar line integral formula:

F of R of T times magnitude of R prime is 2 T times square root of 4 T squared plus 1

The easiest way to evaluate this integral is via u-substitution. Notice that when doing this we also replaced the original bounds of the integral, which are values of t, with the corresponding values of u:

Integral from 0 to 1 of 2 T times square root of 4 T squared plus 1 is 1 sixth times 5 radical 5 minus 1

Piecewise Paths

Does it make sense to evaluate a line integral over a path that is continuous but defined piecewise? For example, to integrate f(x,y) = x + y2 along the quarter of the circle x2 + y2 = 1 that’s in the first quadrant and then along the line segment from (0,1) to (0,2)? If so, how do you calculate the integral?

Path of 2 segments. First is a quarter circle around origin in quadrant 1, second is a line along Y axis from 1 to 2

Such a thing does indeed make sense, and thinking of the line integral as a Riemann sum indicates that you can evaluate the integral as the sum of separate integrals over each part of the path (as long as the parts don’t overlap each other, and together describe the whole path):

Integral over path is integral over first part plus integral over second part and so on

For each part of the path, we can find a separate r(t):

R 1 of T is vector cosine T comma sine T for T between 0 and pi over 2; R 2 is vector 0 comma T for T between 1 and 2

Now use these functions for the parts of the path in separate line integrals, and add the results:

Sum of integrals over 2 parts of path is 10 thirds plus pi over 4

Problem Set

On scalar line integrals, and also double integrals in polar coordinates.

Work on it this week and next, grade it the week after break.

See the handout for details.

Next

Vector fields: vector-valued multivariable functions.

Please read “Examples of Vector Fields” through “Vector Fields in R3” in section 5.1 of the textbook.

Bring Mathematica to class, since we’ll look at how to plot vector fields with it.

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