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:
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!
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 $2\pi$.