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!
best regards
Without integration you can start from the area of a circle wich is given by: $\displaystyle A=\pi \cdot r^2$
The cartesian equation of an ellips is:
$\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=1$
and if $\displaystyle a=b$ then we get a circle with radius $\displaystyle a$. That means the area of an ellips is:
$\displaystyle A=\pi\cdot a\cdot b$, because if $\displaystyle a=b$ (if you've a circle) you get the area of a circle with radius $\displaystyle a$: $\displaystyle A=\pi\cdot a^2$
Is this what you're looking for? ...
Not sure if you are looking for this one:
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 $\displaystyle \Delta x$ and whose length is the appropriate y-value ($\displaystyle = y_C$). Divide the length a into n equal parts (that means $\displaystyle \Delta x = \frac an $) then you get the area of the quarter circle by:
$\displaystyle A_{\tfrac14c} = \lim_{\Delta x \to 0}\left(\sum_{i=1}^n y_C_i \cdot \Delta x\right) = \frac14 \cdot \pi \cdot a^2$
2. An ellipse is produced by a perpendicular dilation (not sure if this is the correct expression) with a fixed quotient:
$\displaystyle \dfrac{y_E}{y_C} = \dfrac ba~\implies~y_E=\dfrac ba \cdot y_C$
3. The area of the quarter ellipse is consequently calculated by:
$\displaystyle A_{\tfrac14e} = \lim_{\Delta x \to 0}\left(\sum_{i=1}^n y_E_i \cdot \Delta x\right) = \lim_{\Delta x \to 0}\left(\sum_{i=1}^n \frac ba \cdot y_C_i \cdot \Delta x\right)=\frac ba \cdot \lim_{\Delta x \to 0}\left(\sum_{i=1}^n y_C_i \cdot \Delta x\right)= \frac ba \cdot \frac14 \cdot \pi \cdot a^2=\frac14 \cdot \pi \cdot b \cdot a$
4. Determine the area of a complete ellipse.