Find the length of the curve. r(t)=4t,t2,t3, 0≤t≤1

Question

Find the length of the curve.  r(t)=4t,t2,t3, 0≤t≤1

Julie Mathew · Accepted Answer

To find the length of the given parametric curve
Length formula for the parametric curve with prescribed limits
If $r (t) = (f (t), g (t), h (t)), a \leq t \leq b$
then, L =length of the parametric curve segment
$= L = \int_{a}^{b} {(f^{'} {(t)}^{2} + g^{'} {(t)}^{2} + h^{'} {(t)}^{2})}^{\frac{1}{2}} d t$
Now, this integral cannot be evaluated in a closed form, as the integrand is not the derivative of a closed expression
Given $r (t) = (4 t, t^{2}, t^{3}), a \leq t \leq b$
$L = \int_{a}^{b} {(f^{'} {(t)}^{2} + g^{'} {(t)}^{2} + h^{'} {(t)}^{2})}^{\frac{1}{2}} d t$
$= \int_{0}^{1} (16 + 4 t^{2} + 9 t^{4})^{\frac{1}{2}} d t$
So, L can be estimated by a suitable numerical procedure, say , Simpson's formula: The approximate value of L is found to be 4.36

Find the length of the curve. r(t)=4t,t^2,t^3,\ 0\leq t\leq1

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