Question

Use Green’s Theorem to evaluate the line integral along the given positively oriented curve.

C is the boundary of the region enclosed by the parabolas

y = x2 and x = y2

Answer 1

Step 1:

The integral is and parabolas are .

To find the point of intersection, equate the parabolas .

Substitute in the parabola .

Substitute in the parabola .

Then the intersection points are .

Step 2:

Greens theorem :

If C  be a positively oriented closed curve, and R be the region bounded by C, M  and N  are the partial derivatives on an open region then

The region bounded by the two parabolas as

.

Solution :

.