I have a line integral (I hope it is) question as:
And the solution to this is :
Now when I try to solve the question using parametric method taking:
$x(t)=t$, $y(t)=2t$
and using the below formula.
$\displaystyle \int _{\mathcal {C}}f(\mathbf {r} )\,ds=\int _{a}^{b}f\left(\mathbf {r} (t)\right)|\mathbf {r} '(t)|\,dt$
I get an answer $33\sqrt{5}$
Can please anyone tell if I am somehow wrong or the solution above contains mistake(s)?
Taking $\mathbf{r}(t)$ as mentioned in your question one has $$\displaystyle \int _{\mathcal {C}}f(\mathbf {r} )\,ds=\int _{a}^{b}f\left(\mathbf {r} (t)\right)|\mathbf {r} '(t)|\,dt = \int_{0}^{1}(4t^3 + 160t^4 ) \cdot \sqrt{5} ~ dt = 33\sqrt{5},$$
where the last equation is due to the solution from your question or simple tedious calculations.