SUNY Geneseo Department of Mathematics
Monday, November 7
Math 223 01
Fall 2022
Prof. Doug Baldwin
The critical-points-and-boundaries part of problem 1 of problem set 9?
Approach this as an absolute extreme values problem, similar to one we did in class. But it’s simpler than the class example, in that the boundary is a triangle with two sides along the positive x and y axes, and a known equation for the third side (so no need to work out the equation from slope and intercept):
Also, from inspection of the function, it will be 0 everywhere along both axes, which takes care of any extreme values on those boundaries or at the endpoints of any boundary. So the only places you need to look for critical points more carefully are in the interior of the triangle and along the hypotenuse.
The Mathematica commands you might find helpful for this problem are being able to define
functions (e.g., to define the production function and maybe the constraint function), and
the Solve command for solving equations or systems of equations. You can
download a notebook with some examples of these things.
Problem set 10, on integration over rectangular and general regions in rectangular coordinates, is ready.
Work on it this week and grade it next week.
See the handout for the details.
…or slightly more accurately, evaluating double integrals over regions defined in polar coordinates.
Based on “Polar Rectangular Regions of Integration” and “General Polar Regions of Integration” in section 4.3 of the textbook.
Evaluate integrals in polar coordinates much as you would in rectangular, but with one important difference: the “dx dy” differentials in rectangular coordinates, which come from the area of a rectangle in a Riemann sum, become “r dr dΘ” in polar, coming from the area of a rectangle in a Riemann sum, but a rectangle whose extent in the Θ direction depends on radial distance as well as the change in angle.
The State University of New West Dakota has decided to consolidate all campus buildings into a single gigantic tower. There will be a glassed-in cafe and greenhouse around part of the base of the tower. In a polar coordinate system with its center at the center of the base of the tower, this glass area extends from Θ = 0 radians to Θ = π/2 radians, and the height of the glass roof is given by h = 144 - r2, for 10 ≤ r ≤ 12. Distances are measured in tens of meters, if you want units (although that makes the roof outrageously high where it meets the wall of the tower). What’s the volume of the cafe/greenhouse?
Find volume by integrating the height of the roof over the area of the cafe. From the way the problem is described, it’s most natural to do this in polar coordinates.
We started thinking about the following problem, and will finish it tomorrow: What is the volume inside the cylinder x2 + y2 = 1 and between the planes z = y + 2 and z = -(x+2) = -x - 2?
The main difference between this and the cafe problem is that this one lends itself to integrating in polar coordinates, because of the cylinder’s circular cross-section, but much of the problem is described in rectangular coordinates. So you need to convert between the two. We’ll try doing that tomorrow.
The slanted cylinder problem.
Time permitting, we’ll also start talking about multiple integrals and mass-related problems.