I have been informed that on March 7th from 6:00am to 6:00pm Central Time Lamar University will be doing some maintenance to replace a faulty UPS component and to do this they will be completely powering down their data center.
Unfortunately, this means that the site will be down during this time. I apologize for any inconvenience this might cause.
Paul
February 18, 2026
Section 16.5 : 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.
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.