Let f be the function given by: f(x)=1/5x^5/2+x^-1/2, for all x in [1,4]. Find the length of the graph of f.
f = 1/5 x^5/2 + x^-1/2
df/dx = 1/2 x^3/2 - 1/2 x^-3/2
ds/dx = sqrt(1+(1/2 x^3/2 - 1/2 x^-3/2)^2)
= 1/2 x^3/2 + 1/2 x^-3/2
s = ∫[1,4] 1/2 x^3/2 + 1/2 x^-3/2 dx
= (1/5 x^5/2 - x^-1/2) [1,4]
= [(32/5 - 1/2)-(1/5 - 1)]
= [59/10 + 8/10]
= 67/10