Answer
$\int_{C}∇ f.dr=6$
Work Step by Step
Given: $x=t^{2}+1$ and $y=t^{3}+t$
Suppose $C$ is a smooth curve. Since, the gradient function is continuous and we know that $f$ is differentiable on $C$.
Apply Fundamental Theorem of line integral.
$\int_{C}∇ f.dr=f(r(1))-f(r(0))$
when $t=1$, we have $x=1^{2}+1=2$ and $y=1^{3}+1=2$
This implies $f(r(1))=f(2,2)$
From the table, we can see that $f(r(1))=f(2,2)=9$
when $t=0$, we have $x=0^{2}+1=1$ and $y=0^{3}+0=0$
This implies $f(r(0))=f(1,0)$
From the table, we can see that $f(r(0))=f(1,0)=3$
Thus, $\int_{C}∇ f.dr=f(r(1))-f(r(0))=9-3=6$
Hence, $\int_{C}∇ f.dr=6$
