What is the arc length of the curve given by r(t)= (1,t,t^2) on t in [0, 1]?
1 Answer
Jun 20, 2018
Explanation:
r(t)=(1,t,t^2)
r'(t)=(0,1,2t)
Arc length is given by:
L=int_0^1sqrt(0^2+1^2+(2t)^2)dt
Simplify:
L=int_0^1sqrt(1+4t^2)dt
Apply the substitution
L=1/2intsec^3thetad theta
This is a known integral. If you do not have it memorized apply integration by parts or look it up in a table of integrals:
L=1/4[secthetatantheta+ln|sectheta+tantheta|]
Reverse the substitution:
L=1/4[2tsqrt(1+4t^2)+ln|2t+sqrt(1+4t^2)|]_0^1
Insert the limits of integration:
L=1/2sqrt5+1/4ln(2+sqrt5)