Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 16 - Vector Calculus - 16.3 The Fundamental Theorem for Line Integrals - 16.3 Exercises - Page 1134: 2

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, $\int_{C}∇ f.dr=6$
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