Consider the hyperbola x^2 - y^2=1.?

Question

What is the arc length with y greater than or equal to 0 from x=2 and x=3?

Answer 1

Well, just recall what arclength is:

S = int(a to b) of "sqrt( 1 + [ y' ]^2)"

Doing algebra:
y = sqrt( x^2 - 1 )
y' = x/sqrt( x^2 -1)

Now, find out what [ y' ]^2 equals:

[ x/sqrt( x^2 -1) ]^2 = x^2/(x^2 - 1)

Plug it in and let your calculator do the rest of the work.

Answer 2

y= sqrt(x^2-1) [This is top half of hyperbola]
y' = x/sqrt(x^2-1) --> (y')^2 = x^2/(x^2-1)
s = integral 2 to 3 sqrt(1-(y')^2) dx
s = integral 2 to 3 sqrt(1-x^2/(x^2-1) dx
s = integral 2 to 3 sqrt(1/(x^2-1)dx
s = ln(x+sqrt(x^2-1)) evaluated from 2 to 3
s = ln(3+sqrt(8)) - ln(2 +sqrt(3)) <--exact solution
s = .445789 <-- approximate solution