I'm not sure if I understand how to parameterize curve correctly. I came across a question which was : Find the value of the integral of a function $g(x,y) = 4x^3 + 10y^4$ along the straight line segment from $(0,0)$ to $(1,2)$ in the $xy$ plane.
I paramatrized using $r(t) = < t, 2t >$, giving me $r'(t) = <1,2>$, the norm of which is $\sqrt5$.
Now if I substitute $x=t$ and $y=2t$ to obtain $g(t, 2t)$ and integrate from $t=0$ to $t=1$ then that integral times norm$=\sqrt5$ should be my answer right?
But the solution in my question book doesn't includes $\sqrt5$, I'd like to know if my procedure is right or not and what should be the correct answer. I'm getting $33\sqrt5 $ while the given answer is $33$.
PS : I'm currently on my phone, I shall format question into TeX later. Thanks for bearing with me until.
You are right, if $S$ is the straight line segment, then
$$\int_Sg(x,y) ds = 33 \sqrt{5}.$$