SUNY Geneseo Department of Mathematics
Friday, April 14
Math 223
Spring 2023
Prof. Doug Baldwin
(No.)
Finishing the questions and examples from Wednesday.
Examples, especially involving triple integrals?
Applications to volume?
Find the integral of x2y + y2z over the “pyramid” (technically a tetrahedron, meaning a triangular pyramid) with vertices at (0, 0, 0), (1, 0, 0), (0, 2, 0), and (0, 0, 2).
Wednesday we figured out what the bounds should be for evaluating this as an iterated integral (see the online notes for how we did it). Our iterated integral is…
This integral involves integrating a series of increasingly complicated polynomials, which doesn’t particularly build new insights about integration, but is painful. So let’s use this as an excuse to learn a bit about iterated integrals in Mathematica.
The function you need is Integrate, which you’ve already seen in connection with integrating vector functions. Now, you give Integrate a scalar function, and multiple bounds, ordered from outer-most integral to inner-most.
We put these ideas into Mathematica, which you can download here to read and experiment with.
As another example, consider integrating f(x, y) = xy2 over the region between the curves x = y2 and x = 4. This provides an example of an integral where it’s easiest to define the region in terms of y rather than x, as well as a few other twists on multiple integrals that we haven’t seen yet.
To find bounds for an iterated integral, start by figuring out where the curves that define the region intersect, i.e., where x = y2 = x = 4, i.e., where y2 = 4. This happens when y = 2 or y = -2. That gives the bounds for the integral with respect to y; bounds for the x integral are the two curves (keeping in mind that y2 is less than 4 in this region, so the y2 curve is the lower bound:
Once we know the bounds, evaluating the integral uses familiar methods:
The n-dimensional volume V inside a closed bounded region R is the integral of 1 over R. In abstract terms, this looks like…
For example, apply this idea to finding the volume of the space between the square 1 ≤ x ≤ 2, 2 ≤ y ≤ 3 in the xy plane and the surface z = 16 - x2 - y2.
Having a concrete region R, we can describe its bounds and turn the abstract triple integral into a specific iterated integral:
Evaluating the iterated integral uses rules we’ve used before:
Problem set 11, on multiple integrals, is available.
Work on it next week, grade it the week after.
Double integrals in polar coordinates.
Please read “Polar Rectangular Regions of Integration” in section 4.3 of the textbook.