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Find an arc length parametrization

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
 
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1 Answer

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.

Comments

i tried solving for t and solving it throuhg wolffram and im still doing something wrong
 
Did you get
 
t= 1/9 ( (9s+64)2/3-16) ?