Consider the path <10t,5t^2,5ln(t)> defined for t>0 . Find the length of the curve between the points (10,5,0) and (20,20,5ln(2)).

Answer 1

Let x(t) = <10t, 5t², 5lnt>

x(1) = <10, 5, 0> and x(2) = <20, 20, 5ln2>

Arc Length = ∫_{(1 to 2)} √[((10t)')² + ((5t²)')² + ((5lnt)')²]dt = ∫_{(1 to 2)}√(100 + 100t² + 25/t²) dt

= ∫_{(1 to 2)} √[(100t⁴ + 100t² + 25) / t²] dt = 5 ∫_{(1 to 2)} √ [(4t⁴ + 4t² + 1) / t²]dt

= 5∫_{(1 to 2)} √[(2t² + 1)² / t²]dt

= 5∫_{(1 to 2)} [(2t²+1) / t)]dt = 5∫_{(1 to 2)} (2t + 1/t)dt ←Evaluate.