Question

Y = (1/4)x^2 - (1/2)lnx..over the interval (1, 7e) ...what is the arc length ?

Answer 1

So, f[x] = 1/4x^2 - 1/2Ln(x)
thus f'[x] = 1/4*2x - 1/2*(1/x) = x/2 - 1/2x
thus f'[x]^2 = (x^2)/4 - 2*(x/2)*(1/2x) + 1/(4x^2) = (x^2)/4 - 1/2 + 1/(4x^2)
thus f'[x]^2 + 1 = (x^2)/4 + 1/2 + 1/(4x^2) = (x/2 + 1/2x)^2
thus Sqrt[...] = (x/2 + 1/2x)