Ellipse or ellipsoid calculations

• equation of ellipse (x/a)^2 + (y/b)^2 =1
• let x' = x/a, and y'=y/b.
• equation in these coordinates is x'^2 + y'2 =1, a circle of radius 1
• area of an ellipse = integral dx dy over the bounding curve
• going to the new coordinates we have area = ab integral dx' dy' over the curve
• since the bounding curve in the new coordinates is a unit circle, the area is obviously pi
• which makes the area of the ellipse pi ab

• ellipsoid calculations go the same way, starting with (x/a)^2 + (y/b)^2 + (z/c)^2 = 1
• when we change to x' = x/a, y' = y/b, and z' = z/c, the bounding surface is a unit sphere
• then the volume of a sphere is easily worked out to be 4/3 pi abc

Calculating integrals like x^2 or xy or y^2 over an ellipse turns out to be trivial once one has transformed to the unit sphere

• Show the integral of x^2 over an ellipsoid transforms to
• a^3bc integral x'^2 over the volume of a unit sphere
• since x' and y' and z' are equivalent over a unit sphere
• integral of x'^2 is 1/3 integral of r'^2 over the unit sphere
• integral of r'^2 (4 pi r'^2 dr') from 0 to 1 is just 4/5 pi
• this makes the integral of x^2 over an ellipsoid 4pi/15 a^3bc.