Question

Calculate the arc length of the indicated portion of the curve r(t).
R(t)=2t^3i +11^3j+10t^3k,1 less than or equal to t less than or equal to 3

Answer 1

The formula for arc length is simply just \[\int\limits_a^b || r'(t) || \ dt\]

It's pretty simple to derive actually, All we're doing is saying that at some point we have a vector on our line and then slightly later we have moved an infinitesimal amount, dt. Just doing simple vector subtraction we get that little segment of the line is their difference, which looks a lot like the derivative! \[r'(t)=\lim_{dx \rightarrow 0}\frac{r(t+dt)-r(t)}{dt}\] So we multiply the derivative by dt to get the length of that really tiny vector. \[r'(t) dt\] So that's just one little tiny segment of the line. But this little segment has a direction and if you were to do a loop-de-loop all the vectors would cancel each other out. So we need to get rid of that by just looking at the length of the derivative. \[||r'(t)|| dt\] So that's the length of ONE itty bitty tiny part of the line. Now we just add them up from point a to point b. \[\int\limits_a^b ||r'(t)|| \ dt\] and there you go. If you want me to explain it more so that you can understand it just ask.