where formula_20 and formula_51 are the lengths of the semi-major and semi-minor axes, respectively. The area formula formula_412 is intuitive: start with a circle of radius formula_51 (so its area is formula_414) and stretch it by a factor formula_415 to make an ellipse. This scales the area by the same factor: formula_416 It is also easy to rigorously prove the area formula using integration as follows. Equation () can be rewritten as formula_417 For formula_418 this curve is the top half of the ellipse. So twice the integral of formula_419 over the interval formula_420 will be the area of the ellipse: