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

1 Answer
Apr 2, 2018

#A=2sqrt2#

Explanation:

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:
enter image source here
#A=sqrt((x_1-x_2)^2+(y_1-y_2)^2)=sqrt(4+4)=sqrt8=sqrt(4*2)=2sqrt2#

This gives us the same result as the integral, showing that either method works, although in this case, I'd recommend the graphical method because it is simpler.