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How would you find the arc length of $r(t) = \langle\sqrt{t}, t,t^2\rangle$ for $1\le t\le 4$? This isn't a homework question, I'm just trying to understand how to properly solve a question such as this. I'm working off the textbook Multivariable Calculus by Stewart, and the solutions manual isn't quite explicit on how to do the hardest bits.

I can solve up to the point where: $r'(t) = \sqrt{(1/4t) + 1 + 4t^2}$ but I'm stuck on how to proceed from this point onwards.

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It doesn't change the difficulty of the problem, but it would be better to have the upper limit something besides $t$ and the derivative of arc length is $s'$, not $r'$ as $r$ has already been used. –  Ross Millikan Sep 25 '12 at 20:53
    
There's a nice example of how to do this at tutorial.math.lamar.edu/Classes/CalcIII/VectorArcLength.aspx –  user22805 Sep 25 '12 at 20:57
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1 Answer

If $r : I \subset \mathbb{R} \to \mathbb{R}^n$ is a curve in $n$-space, we define the arclenght from a initial point to a final point to be the integral of the norm of the tangent vector. That means that the arclength $s : I \subset \mathbb{R} \to \mathbb{R}$ is given by:

$$s(t) = \int_{t_0}^t \| r'(u) \| \,du$$

You've found the tangent vector to the curve, now you have to compute it's norm and then integrate from a starting point to an arbitrary point. Remember that the euclidean norm is given by the square root of the inner product of the vector with itself.

Once you've found the arclength function you can just input the point you want (in this case it's $t = 4$) and you'll be fine. Ah, of course, sometimes you can found an integral that can't be expressed as combination of elementary functions (for instance, when computing the arclength of an ellipse), in that case the best you can do is estimate the integral numerically.

In fact once you've set up the integral and everything else it is just calculating a one dimensional integral as in one variable calculus. The "geometry" of the problem is really understanding and applying the definition of arclength.

If you want to have more background on the topic I recommend reading the first chapter of Differential Geometry of Curves and Surfaces from Manfredo do Carmo, it covers this topics and has very good exercises. There is even one exercise that aims to explain why this definition of arclength is a good definition.

I hope that this aids you somehow.

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