Find the area of the region enclosed by the ellipse x2/a2 + y2/b2 = 1 both
Solution. See Example 2 on page 430 of the text for the solution to 1. For 2, let's label the axes with u and v instead of the usual x and y. Consider the unit circle u2 + v2 = 1, which has area . The transformation, or change of coordinates, given by x = au and y = bv transforms this circle into the ellipse in question:
To determine how this transformation affects areas, consider what happens to the square with corners (0,0), (0,1), (1,0), and (1,1) [in terms of u and v], which has area 1. It is transformed into the rectangle with corners (0,0), (0,b), (a,0), and (a,b) [in terms of x and y], which has area ab. Hence the transformation multiplies area by a factor of ab; it follows that the area of the ellipse is ab.
Bonus. Suppose you draw a number of circles on a blank piece of paper. This divides up the paper into a number of regions whose borders are made up of circular arcs. Prove that you can colour these regions with only colours in such a way that no two regions that have a common border have the same colour.
Solution. Suppose, for the sake of argument, that white and black are the two colours we want to use. Colour the regions as follows: count how many circles each region is inside of, and then colour it white if this number is even and black if this number is odd.
We need to check that two regions with a common border will not end up being the same colour. Observe that if two regions share a common border they are inside all the same circles except for one: one region is inside the circle which forms (part of) the border between them, and the other region is not. Since the number of circles each is inside of must therefore differ by one, one is inside an odd number of circles and the other is inside an even number of circles. It follows that one of the two regions gets coloured white and the other black.