Applet: The line integral over multiple paths of a conservative vector field
The integral of conservative vector field $\dlvf(x,y)=(x,y)$ from $\vc{a}=(3,-3)$ (cyan diamond) to $\vc{b}=(2,4)$ (magenta diamond) doesn't depend on the path. Path $\dlc$ (shown in blue) is a straight line path from $\vc{a}$ to $\vc{b}$. Paths $\adlc$ (in green) and $\sadlc$ (in red) are curvy paths, but they still start at $\vc{a}$ and end at $\vc{b}$. Each path has a colored point on it that you can drag along the path. The corresponding colored lines on the slider indicate the line integral along each curve, starting at the point $\vc{a}$ and ending at the movable point (the integrals alone the highlighted portion of each curve). Moving each point up to $\vc{b}$ gives the total integral along the path, so the corresponding colored line on the slider reaches 1 (the magenta line on the slider). This demonstrates that the integral is 1 independent of the path.
A potential function of $\dlvf(x,y)$ is $f(x,y)=\frac{1}{2}x^2 + \frac{1}{2}y^2$, as $\nabla f = \dlvf$. Therefore, by the gradient theorem, the integral is $f(\vc{b})-f(\vc{a}) = 10-9=1$.
Applet file: line_integral_conservative_vector_field.ggb
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