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I search up this rather common question but I never found an exact answer. For my Calculus 2 class, I need to "set up the integral that represents the arc length of the ellipse and simplify the integrand." Here's what I've done so far: enter image description here

I used parametric equations (not sure if that's the right thing to do), but I have no idea how to substitute the upper and lower limit of the integral. I don't quite understand how the upper/lower limit works(beta and alpha on the picture) since there seems to be two variables here (I didn't learned multivariable yet).

Can someone help me with this assignment? Thanks a lot!

You Xiao Ruan
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The written formula for arc length is correct. Parametrizing the ellipse as you did $$ (x(t),y(t))=(a\cos t,b\sin t) $$ you need $t\in [0,2\pi)$ to plot sketch the curve of the ellipse once (think in analogy to the circle). This gives you your lower bound, 0, and upper bound $2\pi$.

operatorerror
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  • Is there really no way to obtain the upper/lower limit by algebraic way (like U-sub, you change the upper and lower limit for each sub accordingly). – You Xiao Ruan Mar 30 '17 at 02:06
  • @YouXiaoRuan But when you substitute, you have an integral to start with. Here you're trying to come up with an integral. – martin.koeberl Mar 30 '17 at 02:07
  • I got it, but would it be much more complicated if we want to calculate a specific arc length of the ellipse? – You Xiao Ruan Mar 30 '17 at 02:12
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    @YouXiaoRuan there is no closed form expression for arc length on an ellipse. Area is easy, not perimeter. – Will Jagy Mar 30 '17 at 02:14