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

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Answer 1 · 2016-08-24T12:23:33

see below

I'm assuming that the #t# is indeed time, so conveniently for arc length we have

#s = int_3^4 dot s \ dt#

# = int_3^4 abs ( vec v) \ dt#

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

with #vec v = (d vec r)/dt = ((-1),(4t),(-3t^2))#

# implies int_3^4 sqrt(1 + 16t^2 + 9t^4) \ dt approx 39.6 " units"#

the actual integration was done by computer.