Question

How are certain formulæ for areas of circles and ellipses related to calculus?

Answer 1

Area of an Ellispe of the form

`x^2/a^2+y^2/b^2=1`

can be found by `pi cdot a cdot b`, which can be viewed as a general form of the area of a circle since the equation of the circle

`x^2+y^2=r^2 Rightarrow x^2/r^2+y^2/r^2=1`,

which is an ellipse with `r=a=b`; therefore, the area of the circle is `picdot r cdot r=pir^2`.

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I hope that this was helpful.

Answer 2

Unlike the area, the perimeter of an ellipse cannot easily be written down in closed form, like the formula for the circumference of a circle.

Explanation:

There is a whole area of advanced maths dealing with elliptic integrals. Not only does this give the perimeter of an ellipse as an infinite series, but leads on to solutions for the period of a simple pendulum with large amplitude, [such as here].(http://www.rowan.edu/colleges/csm/departments/math/facultystaff/osler/125%20The%20Perimeter%20of%20an%20Ellipse%20as%20in%20Math%20Sci.pdf)