i know that an ellipse area can be easily computed by integration
however i wonder if there might be ageometric way to get the same result?
i tried really hard but all i got were bad approximations
any ideas? thank in advance!
Without integration you can start from the area of a circle wich is given by:
The cartesian equation of an ellips is:
and if then we get a circle with radius . That means the area of an ellips is:
, because if (if you've a circle) you get the area of a circle with radius :
Is this what you're looking for? ...
1. Consider a quarter circle. The area of the quarter circle can be approximated by the sum of areas of small rectangles whose width is and whose length is the appropriate y-value ( ). Divide the length a into n equal parts (that means ) then you get the area of the quarter circle by:
2. An ellipse is produced by a perpendicular dilation (not sure if this is the correct expression) with a fixed quotient:
3. The area of the quarter ellipse is consequently calculated by:
4. Determine the area of a complete ellipse.