ARC LENGTH

 

Homework Help

 

Q. Consider the curve r=(e^(-4t)cos(-7t), e^(-4t)sin(-7t), e^(-4t).   compute the arclength fuction s(t): (with initial point t=0).

 

Ans.

 

r'(t) = (e^(3t) (3 cos(-4t) + 4 sin(-4t)), e^(3t) (3 sin(-4t) - 4cos(-4t)), 3e^(3t))

So, ||r'(t)|| =
= e^(3t) * sqrt[(3 cos(-4t) + 4 sin(-4t))^2 + (3 sin(-4t) - 4cos(-4t))^2 + 3^2]
= e^(3t) * sqrt[9 + 16 + 9], via cos^2(x) + sin^2(x) = 1
= sqrt(34) * e^(3t).

Thus, the arc length function s(t) equals
integral(0 to t) sqrt(34) * e^(3t) dt
= (sqrt(34)/3) * (e^(3t) - 1)

Ref: Image source