2007-04-11 04:53:52 · 6 answers · asked by r wall 3 in Science & Mathematics ➔ Mathematics
The equation of an ellipse is
x^2/a^2+y^2/b^2=1
b^2*x^2+a^2*y^2=a^2*b^2
solve for y^2
y^2=b^2*(a^2-x^2)/a^2
Because of symmetries the
area of an ellipse is
2*Int[b/a*sqr(a^2-x^2)]dx
(x=-a,x=a)=
2*b/a*Int[sqr(a^2-x^2)]dx
(x=-a,x=a)=
put x=asinu and you get
2*b/a*int[a^2*(cosu)^2]du
(u=-pi/2,u=pi/2)
Using for (cosu)^2=(1+cos2u)/2
Finally you get
Area=pi*a*b
2007-04-11 05:29:03 · answer #1 · answered by katsaounisvagelis 5 · 1⤊ 0⤋
Sure ... take half an ellipse, integrate and then double the answer. You'll find out a circle is a degenerate ellipse. The area for an ellipse is Pi x a x b where a and b are the semi minor and major axes. If you look at a circle, the minor axes are both equal to r so you get Pi r^2.
2007-04-11 12:03:59 · answer #2 · answered by Gene 7 · 0⤊ 0⤋
I would integrate 1/4 of the ellipse then quadruple the result. I mean use the center of the major axis and the end of the major axis as the limits of intergration.
You could integrate the top half, that is, use the ends of the major axis as limits but if the ellipse is centered at the origin the result would be zero.
2007-04-11 12:04:07 · answer #3 · answered by bignose68 4 · 0⤊ 0⤋
I suggest integrating over one fourth of the area. Find the equation of the upper portion of the ellipse. Expect to use a trig substitution.
2007-04-11 12:06:29 · answer #4 · answered by fcas80 7 · 0⤊ 0⤋
Yes, it is possible
x^2/a^2+y^2/b^2=1
Find y from this equation.
integrate it from -a to a
2007-04-11 12:08:09 · answer #5 · answered by iyiogrenci 6 · 0⤊ 0⤋
what the hell.
just do it the same way you would find the area of anything else by integration.
2007-04-11 12:02:54 · answer #6 · answered by the misomaniac 3 · 0⤊ 1⤋