The line integral of (2x+9z) ds where the curve is given by the parametric equations x=t, y=t^2, z=t^3 for t between 0 and 1. don't respond if you can't explain how to get out of an impossible square root.

Let r = (t,t^2,t^3)

Then r' = (1, 2t, 3t^2)

General Line integral is:


The limits are 0 to 1
f(r) = 2x + 9z = 2t +9t^3
|r'| is magnitude of derivative vector 



Fortunately, this simplifies nicely with a 'u' substitution.

Let u = 1+4t^2 +9t^4

du = 8t + 36t^3  dt



After integrating using power rule, replace 'u' with function for 't' and evaluate limits: