inkyvoyd
  • inkyvoyd
vince33 Could you please do your homework by yourself, or wolfram alpha? Also, my question is, find the area of a general ellipse in the form with the xy term using integration and trig substitution.
Mathematics
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chestercat
  • chestercat
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dumbcow
  • dumbcow
well area of ellipse = pi*a*b, where a and b are major and minor axis
inkyvoyd
  • inkyvoyd
Hmm. Can you show your work please?
dumbcow
  • dumbcow
you want area in terms of x,y ?

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inkyvoyd
  • inkyvoyd
Actually, scratch that. I need a hint as to how to transform an ellipse into one with no rotation.
dumbcow
  • dumbcow
http://en.wikipedia.org/wiki/Rotation_%28mathematics%29#Matrix_algebra
inkyvoyd
  • inkyvoyd
Wait, what's the equation of a rotated ellipse?
dumbcow
  • dumbcow
to answer prev question regarding proof of area the equation for general ellipse solved for y \[y = \pm \frac{b}{a}\sqrt{a^{2}-x^{2}}\] due to symmetry, we can integrate to get area in 1st quadrant, then multiply by 4 \[A = 4\frac{b}{a}\int\limits_{0}^{a}\sqrt{a^{2}-x^{2}} dx\] set x = asin(u) dx = acos(u)
inkyvoyd
  • inkyvoyd
dumb, I got the integration, don't spoil it xD
dumbcow
  • dumbcow
haha ok the equation for rotated ellipse, you replace x and y with x' and y' based on formulas which depend on angle of rotation
inkyvoyd
  • inkyvoyd
but the problem is I want the general area given in terms of coefficients. I wanted to do a translation and rotation to make it at the origin, then integrate for area. Is that possible>
dumbcow
  • dumbcow
so you are starting with a slanted ellipse and want the area in terms of coefficients. yes you would have to rotate it so its standard ellipse on xy axis then just determine what a and b are
inkyvoyd
  • inkyvoyd
yes. But how would I know what the general equation of the slanted ellipse was?
dumbcow
  • dumbcow
oh its not given?
dumbcow
  • dumbcow
its usually of the form \[\frac{(x \pm y)^{2}}{c}+\frac{(x \pm y)^{2}}{d} = 1\] but it depends on the angle
dumbcow
  • dumbcow
there probably is a general form for slanted ellipse in terms of theta but i don;t know what it is
dumbcow
  • dumbcow
here we go http://en.wikipedia.org/wiki/Ellipse#General_ellipse
dumbcow
  • dumbcow
so from that general equation, replace x and y with x' and y' solve for y' and integrate are we given the angle? or will the area be in terms of theta? this could get messy

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