Consider the path r(t)=(10t,5t2,5lnt) defined for t> 0. find the length of the curve between the points (10,5,0) and (30,45,5ln(

The length of the curve is of: units.

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The length of a parametric curve in an interval [a,b] is given by:

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The curve is:

Thus:

Between points and

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The length is:

The length of the curve is of units.

A similar problem is given at link

The length of the curve is

We have , but integrating over [1, 3] makes , so . Under the square root, we can factorize

and for the same reason as before,

Then the integral is

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