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Section 5-5 : Fundamental Theorem for Line Integrals

4. Evaluate \( \displaystyle \int\limits_{C}{{\nabla f\centerdot d\vec r}}\) where \(f\left( {x,y} \right) = {{\bf{e}}^{x\,y}} - {x^2} + {y^3}\) and \(C\) is the curve shown below.

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This problem is much simpler than it appears at first. We do not need to compute 3 different line integrals (one for each curve in the sketch).

All we need to do is notice that we are doing a line integral for a gradient vector function and so we can use the Fundamental Theorem for Line Integrals to do this problem.

Using the Fundamental Theorem to evaluate the integral gives the following,

\[\begin{align*}\int\limits_{C}{{\nabla f\centerdot d\vec r}} &= f\left( {{\rm{end point}}} \right) - f\left( {{\rm{start point}}} \right)\\ & = f\left( {0, - 2} \right) - f\left( { - 2,0} \right)\\ & = - 7 - \left( { - 3} \right) = \require{bbox} \bbox[2pt,border:1px solid black]{{ - 4}}\end{align*}\]

Remember that all the Fundamental Theorem requires is the starting and ending point of the curve and the function used to generate the gradient vector field.