Line integral homework

Line Integrals – Part I – In this section we will start off with a quick review of parameterizing curves. This is a skill that will be required in a great many of the line integrals we evaluate and so needs to be understood.

Evaluation of line integrals over piecewise smooth curves is a relatively simple thing to do. All we do is evaluate the line integral over each of the pieces and then add them up. The line integral for some function over the above piecewise curve would be,

Using Green's theorem evaluate integral_c (x^2 - y^2) dx + (2y - x) dy, where c consists of the boundary of the region in the first quadrant that is bounded by the graphs of y = x^2 and y = x^3. Let F=4xyi+9y^2j be a vector field in the plane, and C the path y=6x^2 joining (0,0) to (1,6) in the plane.

Here is the parameterization of the curve. for 0 ≤ t ≤ 1 0 ≤ t ≤ 1. Remember that we are switching the direction of the curve and this will also change the parameterization so we can make sure that we start/end at the proper point.