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

What is the arclength of #f(t) = (t-7,t+7)# on #t in [1,3]#?

Answer 1

Ezra Adams

# 2sqrt( 2 ) #

The Arc Length for a Parametric Curve is given by

# L=int_alpha^beta sqrt((dx/dt)^2+(dy/dt)^2) \ dt #

So in this problem we have (using the product rule):

# x= t-7 => dx/dt = 1#
# y= t+7 => dy/dt = 1 #

So the Arc Length is;

# L= int_1^3 sqrt( (1)^2 + (1)^2) \ dt #
# \ \ = int_1^3 sqrt( 2 ) \ dt #
# \ \ = sqrt( 2 ) int_1^3 \ dt #
# \ \ = sqrt( 2 ) [t]_1^3 #
# \ \ = sqrt( 2 ) (3-1) #
# \ \ = 2sqrt( 2 ) #

Additionally, If we look at the actual graph of the parametric curve:

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