What is the value of the line integral of $f(x,y)$?
The formula that I was taught was line integral = integral (from a to b) f(r(t)||r'(t)||dt
where r is the parameterization of the curve $C$ and ||r'(t)|| is the norm of r'(t).
I've done examples where the curve $C$ is usually a circular curve, in which we can parameterize r by applying $x=rcost$ and $y=rsint$, but im not sure how to parameterize in this case, so that I can apply the formula stated above.
On
The line integral of a scalar function is defined as $$\int_\gamma f\ ds:=\int_{a}^{b}f\left(\mathbf{\vec\gamma} (t)\right)\Vert{\vec\gamma\ ^{'}(t)}\Vert dt$$
In this case our curve is given by the parametric equations
$ \gamma(t)= \begin{cases} x=t \\ y=2{t^2} \end{cases} $
with $t\in \left[0, 1\right]$. So the integral is given by
$$\int_\gamma f\ ds=\int_{0}^{1}\sqrt{16t^2+1}\sqrt{1+16t^2}\ dt=\int_{0}^{1}(16t^2+1)\ dt=\frac{19}{3}.$$
If the curve $C$ is $y=2x^2$ where $0 \leq x \leq 1$, then a parametrization would be
$$x=t, y = 2t^2$$
where $t \in [0,1]$.
Now you can apply the formula that is given to you and solve the problem.