Find an arc length parametrization

Question

Find an arc length parametrization  of r(t)=/<6t^2 ,3t^3.>

 

r'(t)= <12t,9t^2>

length of r'(t)= sqrt(144t^2+81t^4)

integral sqrt(144t^2+81t^4) from 0 to t is t*sqrt(144+81t^2)

u= 144+81t^2

du= 162t dt

1/162 * du= t dt

1/162 integral  from 0 to t = u^(1/2) = 1/162(2/3)u^(3/2) from 144 to 144+81t^2

=1/243((144+81t^2)^(3/2)-64^(3/2)

were do i go to find s and the value t equals to put back in to the orginal

 

Answer 1

Everything so far looks right, except the very last expression should be

 

s(t) = 1/243((144+81t^2)^(3/2)-144^(3/2)).

 

Now solve this expression for t(s) and substitute this for t into the original parametrization r(t). That will be the arc length parametrization.