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Section 16.5 : Fundamental Theorem for Line Integrals

3. Given that $$\displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}$$ is independent of path compute $$\displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}$$ where $$C$$ is the ellipse given by $$\displaystyle \frac{{{{\left( {x - 5} \right)}^2}}}{4} + \frac{{{y^2}}}{9} = 1$$ with the counter clockwise rotation.

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At first glance this problem seems to be impossible since the vector field isn’t even given for the problem. However, it’s actually quite simple and the vector field is not needed to do the problem.

There are two important things to note in the problem statement.

First, and somewhat more importantly, we are told in the problem statement that the integral is independent of path.

Second, we are told that the curve, C, is the full ellipse. It isn’t the fact that $$C$$ is an ellipse that is important. What is important is the fact that $$C$$ is a closed curve.

Now all we need to do is use Fact 4 from the notes. This tells us that the value of a line integral of this type around a closed path will be zero if the integral is independent of path. Therefore,

$\require{bbox} \bbox[2pt,border:1px solid black]{{\int\limits_{C}{{\vec F\centerdot d\vec r}} = 0}}$