SUNY Geneseo Department of Mathematics
Monday, April 10
Math 223
Spring 2023
Prof. Doug Baldwin
In problem set 9, question 2 deals with a 2-variable function but provides 3 coordinates for some points because it’s giving x, y, and f(x,y).
Two suggestions from the mid-semester feedback survey:
Work through example problems in class before asking you to do similar problems in class.
I have mixed feelings on this. When there’s a massive request for examples during the “key ideas and questions” part of class, it does make sense for me to work through a complete example for you. But in most cases, I believe you can get a lot from trying an example problem before seeing its solution, even if you don’t think you can solve it completely: you get a chance to identify how the problem matches examples or ideas in the textbook, to identify specifically what it is that keeps you from making progress, and doing these kinds of things even without solving the problem gives you an important foundation from which to understand the solution when it is presented. So I’ll also keep asking you to think about examples without knowing how to solve them.
Have some “leniency” in evaluating solutions to oral exam questions.
I already try to do this by not grading the work you do in response to my questions during a meeting for things outside this course. If you struggle with those things, I’ll help you get better at them without penalizing you. What I will grade on is how well you understand things that are part of this course. The standard is stricter for problems you’re supposed to prepare written solutions to before a meeting though, since you have time to proofread, think over, get advice on, etc. those problems. I’ll try to be more explicit about what parts of a meeting affect your grades and what don’t though.
Base on “Properties of Double Integrals,” and “Iterated Integrals” in section 4.1 of the textbook, and “Integrable Functions of Three Variables” in section 4.4.
The order in which you evaluate iterated integrals doesn’t matter (Fubini’s Theorem). There’s an interesting similarity here to the fact that it doesn’t matter what order you take derivatives in mixed higher order partial derivatives.
Triple integrals have a formal definition as limits of sums, and it’s an extension to 3 dimensions of the definition of double integrals.
The 6 properties of double integrals (see the book for the complete list).
Generally, ideas from 1- and 2-dimensional integrals extend to 3 (and more) dimensional integrals; this includes the key ideas above.
What’s the volume under one bump in my “egg carton” function, e(x, y) = sin x cos y?
Start by figuring out what region to integrate over. Intuitively, one “bump” covers an interval in x that’s long enough to take sin x from 0 up to 1 and back down to 0, and similarly an interval in y that takes cos y from 0 to 1 and back to 0. The simplest intervals that do this are 0 ≤ x ≤ π and -π/2 ≤ y ≤ π/2:
With these intervals settled, we can set up the integral, and then evaluate it by integrating from the inside out:
As an example of using properties of multiple integrals, consider integrating f(x, y) = x y2. This function is a product of two terms, one of which involves only x and the other only y. So you can split the integral into a product of 2 integrals, each involving only 1 variable. This integral is perhaps simpler to evaluate than the original iterated integral.
You’re getting good now at integrating multivariable functions over regions whose bounds are constants and don’t depend on each other. But what about more complicated regions, i.e., multiple integrals over non-rectangular regions?
Please read “General Regions of Integration” and “Double Integrals over Non-rectangular Regions” in section 4.2 of the textbook to start on this subject.