(922, #3) Evaluate the line integral

where C consists of the arc of the parabola from ( 0, 0 ) to ( 1, 1 ) followed by the line segment from ( 1, 1 ) to ( 0, 0 ), by two methods: a) directly and b) using Green's Theorem.

Solution: C is a simple closed curve:

a)

We break up C into 2 parts.

Green line, : Red line, ,

=

We evaluate an integral over and one over , then add the answers together to get

b)

by C.A.H.