Find the arc length

Question

Find the arc length of the curve c(t)=(t^3/2,cos(2t),sin(2t)), 0<=t<=1.

Answer 1

The individual components of c(t) give the (x,y,z) positions of the curve in space. If you differentiate each component with respect to t, you'll get a "velocity" vector function:

 

v(t)= dc/dt = [(3/2)t^1/2, -2sin(2t), 2cos(2t)].

 

When we compute the arc length of a parametric curve, we want to integrate its speed over the entire parameter interval. (Integrating speed will give us a length or distance. Integrating velocity would give us a displacement instead, and we're not interested in that.)

 

Speed is the magnitude of the velocity vector, so we need to take the square root of the sum of the squares of each component:

 

speed= √[(9/2)t + 4 sin²(2t) + 4 cos²(2t)] = √[(9/2)t + 4].

 

Arc Length= ∫₀¹ √[(9/2)t+4] dt, which can be completed with a simple u-substitution.