y=(1/3)(x^2+2)^(3/2)
dy/dx=(1/3)(3/2)(x^2+2)^(1/2)=xsqrt(x^2+2)
If a tiny displacement from point P(x,y) is dx and dy, the corresponding displacement in the line segment is ds=sqrt(dx^2+dy^2) (Pythagoras), so ds/dx=sqrt(1+(dy/dx)^2) and ds=sqrt(1+(dy/dx)^2)dx=sqrt(1+x^2(x^2+2))dx=sqrt(x^4+2x^2+1)=x^2+1. s=S[0,3]((x^2+1)dx) where S means integral and [] contain the limits, low to high. s=(x^3/3+x)[0,3]=9+3=12.