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Section 5-5 : Fundamental Theorem for Line Integrals

1. Evaluate \( \displaystyle \int\limits_{C}{{\nabla f\centerdot d\vec r}}\) where \(f\left( {x,y} \right) = {x^3}\left( {3 - {y^2}} \right) + 4y\) and \(C\) is given by \(\vec r\left( t \right) = \left\langle {3 - {t^2},5 - t} \right\rangle \) with \( - 2 \le t \le 3\).

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Start Solution

There really isn’t all that much to do with this problem. We are integrating over a gradient vector field and so the integral is set up to use the Fundamental Theorem for Line Integrals.

To do that we’ll need the following two “points”.

\[\vec r\left( { - 2} \right) = \left\langle { - 1,7} \right\rangle \hspace{0.25in}\hspace{0.25in}\vec r\left( 3 \right) = \left\langle { - 6,2} \right\rangle \]

Remember that we are thinking of these as the position vector representations of the points \(\left( { - 1,7} \right)\) and \(\left( { - 6,2} \right)\) respectively.

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Now simply apply the Fundamental Theorem to evaluate the integral.

\[ \displaystyle \int\limits_{C}{{\nabla f\centerdot d\vec r}} = f\left( {\vec r\left( 3 \right)} \right) - f\left( {\vec r\left( { - 2} \right)} \right) = f\left( { - 6,2} \right) - f\left( { - 1,7} \right) = 224 - 74 = \require{bbox} \bbox[2pt,border:1px solid black]{{150}}\]