Math Quiz 9

[GeeTwo]

Originally Posted by MechEng83

The problem gets to the 4ʃʃ(1-x)(1-y)√(x^2+y^2)dxdy GeeTwo derived, but then it does a polar coordinate substitution to make the integration "easier"
It's another approach which gives a closed form solution, demonstrating that there can be multiple ways to validly solve a problem.

This is the route I was following. (I have not looked at the video, so perhaps this is exactly what is there.) My original thought was to substitute r for √(x²+y²), rcosθ for x, and rsinθ for y. dxdy then becomes rdrdθ. While driving to mom's house for Sunday dinner, I realized that I could take advantage of the symmetry of x and y by rotating θ by π/4, so that y is √2(cosθ + sinθ)r/2 and x is √2(cosθ - sinθ)r/2. Then the integration is over an interval symmetric over θ=0, and any odd terms in θ can be tossed (I already figured out while driving that there aren't any, however), and then the integration can be done from 0 to π/4. The fun part is the limit of integration - for r<=1, there's no issue. For 1<r<√2, the limits of one integration need to be set so that the maximum of r is √2/(cosθ + sinθ) (ignoring the required absolute values because I've limited θ to the first quadrant). Alternately, the limits of θ can be set based on r.

Originally Posted by Caleb Sykes

There are uncountably infinite lines of length 0.01, just as there are an uncountably infinite number of lines of length 0.52.

It makes no logical sense to say that there are "more" or "less" of one uncountably infinite thing than another uncountably infinite thing.

It makes just as much sense as to ask "What fraction of integers are even?" There are uncountably many even integers and uncountably many integers, and it is possible to identify a different even integer for every integer (e=2i), so that the sets have the same number of elements. This does not change the equally meaningful statement that exactly half of the integers are even.