(906, #7) Evaluate the line integral:

where C consists of line segments from ( 0, 0 ) to ( 2, 0 ) and from ( 2, 0 ) to ( 3, 2 ).

Solution: The path of integration consists of two line segments:

We split it into parts, C₁ and C_2. C₁is the horizontal part and, on C₁,

x = 2 t
y = 0
t = 0..1

On C₂ , < x, y> = <2, 0> + <1,2> t, so

x = 2 +t
y = 2t
t = 0..1

Then, on C₁

for x y = 0 and dy = 0.

On C₂