Question

Reparametrize the curve with respect to arc length measured from the point where t = 0 in the direction of increasing t. (Enter your answer in terms of s.) r(t) = e^5t cos(5t) i + 5 j + e^5t sin(5t) k r(t(s)) =

          Reparametrize the curve with respect to arc length measured from the point where t = 0 in the direction of increasing t. (Enter your answer in terms of s.) r(t) = e^5t cos(5t) i + 5 j + e^5t sin(5t) k
r(t(s)) =
        

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Calculus: Early Transcendentals
Calculus: Early Transcendentals
James Stewart 8th Edition
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Reparametrize the curve with respect to arc length measured from the point where t = 0 in the direction of increasing t. (Enter your answer in terms of s.) r(t) = e^5t cos(5t) i + 5 j + e^5t sin(5t) k r(t(s)) =
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Transcript

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00:01 Hello friends in this question, let us evaluate the length of the curve using the following formula in some positive interval of 0 to t.
00:07 So s is equal to integral 0 to t x dash of t1 the whole square plus y dash of t1 the whole square plus z dash of t1 the whole square root into d t so this integral we have as integral 0 .0.
00:30 To t we get 3e power 3 t 1 cost 3 t 1 minus 3 e power 3 t 1 sign 3 t 1 the whole square plus 0 square plus 3 e power 3 t 1 sign 3 t 1 plus 3 e power 3 t 1 cause 3 t 1 plus 3 e power 3 t 1 cause 3 3 t 1 t 1 the whole square the whole square root into d t 1 so this integral is equal to 3 root 2 integral 0 to t root of e power 6 t 1 into d t 1 where we integrate this and we write 3 root 2 integral 0 to t 3 power 3 t 1 d t 1 is equal to e power 3t minus 1 into root 2.
01:45 Hence s is equal to e power 3t minus 1 into root 2.
01:51 So this implies that s by root 2 plus 1 is equal to e power 3t where it implies log of s by root 2 is equal to root 2 plus 1 is equal to 3t which implies 0...
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