SUNY Geneseo Department of Mathematics
Wednesday, April 12
Math 223
Spring 2023
Prof. Doug Baldwin
Why did the example yesterday with a substitution in a definite integral change the bounds of the integral?
Because when you do a u substitution in a definite integral, you can (and should) recalculate the bounds to be values of the new variable u instead of values of the old variable x:
An example similar to the homework 9’s problem 2C, about finding the tangent to a level curve by finding a parametric form for the level curve?
Let’s try it for the function f(x, y) = x2/4 + y2/9 and the level curve for f(x, y) = 1:
The implicit equation for the level curve is that of an ellipse. Furthermore, the ellipse extends ±2 units from the origin in the x direction and ±3 units in the y direction:
A good parametric form for ellipses is x = a cos t, y = b sin t, where a and b are the ellipse’s extents in the x and y directions. The tangent to this parametric form, and thus to the level curve, is given by its derivative:
Finally, you could plug in specific t values to get the tangent at particular points. In the problem set, you need to find a value of t that makes the parametric form of the level curve have specific x and y values, and then plug that t into the derivative to get the tangent at that point.
The setting for the problem is finding the minimum distance from a surface to a point. Specifically, the surface is the paraboloid z = x2 + y2, and the point is (0, 0, 3). If you imagine moving up the side of the paraboloid from the origin, the distance to that point will decrease from 3 to some minimum, and then start increasing again to infinity. Because point (0, 0, 3) is on the paraboloid’s axis, there will be circles of points that all have the same distance from (0, 0, 3):
This problem is a constrained optimization problem, where the objective function (the function to be minimized) is distance from point (x, y, z) to (0, 0, 3), and the constraint is that (x, y, z) lies on the parabloid:
With the objective function and constraint identified, the problem is in a form where you could solve it using Lagrange multipliers, which is what I’d suggest.
GREAT Day is is an annual day for Geneseo to showcase students’ research and artistic work.
It’s April 26, 2 weeks from today.
No classes (including ours) that day, to allow students time to present and attend talks, performances, etc.
In that vein, I’m discouraging appointments that day, but won’t outright refuse them.
Including material from “Changing the Order of Integration” in section 4.2, and “Triple Integrals over a General Bounded Region” in 4.4.
You can change the order of the integrals in problems with general bounds, but you also have to correspondingly change the bounds.
Having higher numbers of dimensions means you can have more variables in the bounds of the innermost integrals.
An example that involves setting up bounds and then evaluating an integral in 3 dimensions:
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). Notice that this region is bounded by the xy, xz, and yz planes, and the plane 2x + y + z = 2 (that happening to be an equation for a plane through points (1, 0, 0), (0, 2, 0) and (0, 0, 2)):
To find bounds for the integrals, pick some variable to start with. We decided to start with z, which ranges from 0 to 2. Now think how the range of values for one of the other variables — we chose y — depends on z. The maximum range of y happens when x = 0, and has y ranging up from 0 until it hits the top plane, which happens when y = 2 - z.
Similarly, for given z and y values, x can range up from 0 until it hits the plane. This thinking gives bounds for all 3 variables:
Finally, plug the bounds into iterated integrals to get the integral we’ll need to evaluate:
We’ll evaluate this integral Friday. It’s messy enough to evaluate that Mathematica offers real advantages over doing it by hand, so we’ll also use this as an excuse to look at how Mathematica supports multiple integrals. Bring your computer if you want to try it as we talk.
More examples of multiple integrals and general regions, particularly as applied to volume and area calculations.
No new reading.