What is the arc length of the curve given by #r(t)= (e^tsqrt(2t),e^t,e^(-2t))# on # t in [3,4]#?

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Answer 1 · 2016-08-25T07:52:26

#approx 110.75#

#vec r(t)= ((e^tsqrt(2t)),(e^t),(e^(-2t)))#

for arclength: #s = int_3^4 dot s \ dt#

# = int_3^4 sqrt( vec v * vec v ) \ dt#

#vec v = (d vecr )/ (dt) = ((e^tsqrt(2t) + (e^t )/sqrt(2t)),(e^t),(-2e^(-2t)))#

# implies int_3^4 sqrt( (e^tsqrt(2t) + (e^t )/sqrt(2t))^2 + (e^(t))^2 + (-2e^(-2t))^2) \ dt#

# = int_3^4 sqrt( e^(2t)(2t) +2e^(2t) + (e^(2t) )/(2t) + e^(2t) + 4e^(-4t)) \ dt#

# = int_3^4 sqrt( e^(2t)(2t + 3 + 1/(2t)) + 4e^(-4t)) \ dt#

#approx 110.75# by computer