Home > Calculus > Determining the Length of a Parametric Curve (Parametric Form)

What is the arc length of #r(t)=(t,t,t)# on #tin [1,2]#?

Answer 1

Ezra Adams

#sqrt(3) #

We seek the arc length of the vector function:

# bb( ul r(t)) = << t,t,t >> # for #t in [1,2]#

Which we can readily evaluated using:

# L = int_alpha^beta \ || bb( ul (r')(t)) || \ dt #

So we calculate the derivative, # bb( ul (r ')(t)) #:

# bb( ul r'(t)) = << 1,1,1 >> #

Thus we gain the arc length:

# L = int_1^2 \ || <<1,1,1>> || \ dt #

# \ \ = int_1^2 \ sqrt(1^1+1^2+1^2) \ dt #

# \ \ = int_1^2 \ sqrt(3) \ dt #

# \ \ = [sqrt(3)t]_1^2 #

# \ \ = sqrt(3)(2-1) #

# \ \ = sqrt(3) #

This trivial result should come as no surprise as the given original equation is that of a straight line.

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