# Calculus 3 : Green's Theorem

## Example Questions

### Example Question #3 : Line Integrals

Use Green's Theorem to evaluate , where  is a triangle with vertices  with positive orientation.

Explanation:

First we need to make sure that the conditions for Green's Theorem are met.

The conditions are met because it is positively oriented, piecewise smooth, simple, and closed under the region (see below).

In this particular case , and , where , and  refer to .

We know from Green's Theorem that

So lets find the partial derivatives.

### Example Question #4 : Line Integrals

Use Green's Theorem to evaluate the line integral

over the region R, described by connecting the points , orientated clockwise.

Explanation:

Using Green's theorem

since the region is oriented clockwise, we would have

which gives us

### Example Question #5 : Line Integrals

Use Greens Theorem to evaluate the line integral

over the region connecting the points  oriented clockwise

Explanation:

Using Green's theorem

Since the region is oriented clockwise