How do you find the arc length of the curve y=x^2/2 over the interval [0, 1]?

1 Answer
Sep 5, 2016

approx 2.96 units

Explanation:

s = int_(0)^(2) sqrt(1+(y')^2) \ dx

= int_(0)^(2) sqrt(1+x^2) \ dx

= 1/2[ (x sqrt(x^2+1) +sinh^(-1)(x)) ]_(0)^(2)

= (sqrt(5)+1/2 sinh^(-1)(2) ) units

approx 2.96 units

Computer used for integration and numerical solution

[The basic integral, int sqrt(1+x^2) \ dx...

... can be approached, using the identity cosh^2 z - sinh^2 z = 1...

....so by the sub x = sinh z, dx = cosh z \ dz

... and then maybe a hyperbolic double angle formula]