Greens theorem. find the max value of the line integral where f=(13x^2y+3y^3-y)i-12x^3j and c is any positively oriented closed curve. max=?

The line integral is given by



By Green's theorem, the line integral along is equivalent to the double integral over (the region bounded by )





Now consider the function . We can think of the double integral above as a volume integral; namely, it's the volume of the region below and above the region in the - plane (i.e. ). This volume will be maximized if is taken to be the intersection of with the plane, which means is the ellipse .

For the double integral, we can convert to an augmented system of polar coordinates using



where and . We have the Jacobian determinant




So the double integral, upon converting to our polar coordinates, is equivalent to