I dislike finding arc lengths because so very frequently you end up with a radical you can't easily do away with. Such is my situation in two problems on which I am stuck, and after 1 hour of cogitation, I turn to the Internet to see if you might have a hint on where to go next.

1.) I am given y = (x^3 / 3) + (1 / 4x) for 1<or=x<or=2 and am asked to "Graph the curve and find its exact length" I've got it graphed, but the length? Not so much. I understand that the formula for finding arc length calls for the derivative of the function, so I dutifully dervive the given and arrive at:

y' = x^2 - 4x^-2. Spiffy.

So I set L = Integral from 1-->2 of Root(1 + (x^2 - 4x^-2)^2)dx

Now what? O.o I don't know how to wrestle with this beast. I don't see a u-substitution that readily presents itself and integration by part would just leave me with a REALLY ugly du that I'd be stuck trying to integrate anyway (yuck). If I expand the (x^2 - 4x^-2)^2 I get Root(1+ x^4-8x^-2 + 16x^-4) dx, which doesn't appear to be any better.

I'm open to suggestions!

2.) This time I am given the same instructions as above: Graph the curve and find its length. But here they give me x= (e^t)+(e^-t), y= 5-2t, 0<or=t<or=3

Again, I do a Yeoman's job deriving x and y. x'=(e^t)-(e^-t) and y' = -2.

I then have L = integral from 0-->3 of Root((-2)^2 + ((e^t)-(e^-t))^2)dt. I recognize that e^t-e^-t = 2sinh t, but I don't think that getting into hyperbolic trig is actually simplifying things at all. So I'm stuck here again. Again, I don't see a good u-substitution to use that would make things simple. I mean I could do u = t, but that's just swapping out letters and not simplifying my life any. Any other u value seems to leave dangling bits that would be messy. Integration by part would give me just a hideous looking integral. I don't know of an easy way (or at least a rational, logical way) forward on this one either.

Please help!!!

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