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Calc. 3 problems, Help!!!?
1.) Arc length of <3cos(2t), 3sin(2t), 5t> from 0 <= t <= Pi.
2.) Find the arc length parametrization of r(t) = <3t^(2)-1, 2-t^(2), 5>.
Please give details.
Thank you
1 Answer
- kbLv 79 years agoFavorite Answer
1) r(t) = <3 cos(2t), 3 sin(2t), 5t>
==> r'(t) = <-6 sin(2t), 6 cos(2t), 5>
==> ||r'(t)|| = √(6^2 + 5^2) = √61.
So, the arc length equals
∫(t = 0 to π) √61 dt = π√61.
---------------
2) s(t) = ∫(x = 0 to t) ||r'(x)|| dx
..........= ∫(x = 0 to t) ||<6x, -2x, 0>|| dx
..........= ∫(x = 0 to t) 2x√10 dx
..........= t^2 √10.
Solve for t:
t = (s/√10)^(1/2).
Hence, the arc length parameterization is
r(s) = <3s/√10 - 1, 2 - s/√10, 5>.
I hope this helps!
