# Chapter 16 - Section 16.3 - The Fundamental Theorem for Line Integrals - 16.3 Exercise - Page 1094: 2

$\int_{C}∇ f.dr=6$

#### 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, $\int_{C}∇ f.dr=6$

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