Final answer:
The question involves evaluating a complex contour integral using parameterization, reducing it to a single-variable integral, and interpreting the integral's result as areas under a curve.
Step-by-step explanation:
The student's question is about evaluating a complex contour integral using parametrizations of the contours. For the integral ∫_Γ (z+√z) dz, where Γ is a specific contour path, one would parametrize this path in terms of a single variable to simplify the integral to a form that can be readily evaluated. For instance, if Γ is a path along the real axis, one might choose a parameter t such that z(t) = t, and then express both z and √z in terms of t, which would reduce the integral to a single-variable integral. The next step would be to actually perform the integral, which generally involves finding the antiderivative of the integrand and evaluating it at the bounds defined by the contour.
Furthermore, it is crucial to note the resemblance between the concept of work done by a force and the area under the curve, as explained using Equation 7.5. In the context of such integrals, 'areas' above the x-axis are considered positive, and those below are negative, which is important for interpreting the results of the integral correctly.