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

What is the arclength of #(t-3,t+4)# on #t in [2,4]#?

Answer 1

Emilia Aguilar

#A=2sqrt2#

The formula for parametric arc length is:
#A=int_a^b sqrt((dx/dt)^2+(dy/dt)^2)\ dt#

We begin by finding the two derivatives:

#dx/dt=1# and #dy/dt=1#

This gives that the arc length is:
#A=int_2^4sqrt(1^2+1^2)\ dt=int_2^4sqrt2\ dt=[sqrt2t]_2^4=4sqrt2-2sqrt2=2sqrt2#

In fact, since the parametric function is so simple (it is a straight line), we don't even need the integral formula. If we plot the function in a graph, we can just use the regular distance formula:

By signing up, you agree to our Terms of Service and Privacy Policy

Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.