What's the arc length for the curve?

Question

What's the arc length for the curve defined by r(t)=3sin(t)i+3cos(t)j+4tk for 0<=t<=10? (Answer: 50)

r'(t)=<3cost, -3sint, 4>

 

Answer 1

You already computed v(t) = r'(t) so you need to use the formula for the arc-length.

It will be

∫₀¹⁰ Norm(v(t)) dt

= ∫₀¹⁰ √[(3 cos t)² + (-3 sin t)² + 4²] dt

= ∫₀¹⁰ √(9 cos² t + 9 sin² t + 16) dt

= ∫₀¹⁰ √(9+16) dt

= ∫₀¹⁰ 5 dt

= 5t |₀¹⁰

= 50

Note that I used the identity sin² t + cos² t = 1 which you should know at this point.