Let vector f = (6 x^2 y + 2 y^3 + 4 e^x) i + (7 e^{y^2} + 54 x) j . consider the line integral of vector f around the circle of radius a, centered at the origin and traversed counterclockwise.
(a) find the line integral for a=1 .
(b) for which value of a is the line integral a maximum?

Denote the circle of radius by . is simple and closed, so by Green's theorem the line integral reduces to a double integral over the interior of (call it ):

is a circle of radius , so we can write the double integral in polar coordinates as

a. For , we have

b. Let denote the integral with unknown parameter ,

By the fundamental theorem of calculus,

has critical points when

If , then line integral is 0, so we ignore that critical point. For the other two, we would find .

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