# Green’s Theorem Examples 1

Recall from the Green’s Theorem page that if $D$ is a regular closed region in the $xy$-plane and the boundary of $D$ is a positively oriented, piecewise smooth, simple, and closed curve, $C$ and if $\mathbb{F} (x, y) = P(x, y) \vec{i} + Q(x, y) \vec{j}$ is a smooth vector field on $\mathbb{R}^2$ then Green’s theorem says that:

(1)

We will now look at some examples of applying Green’s theorem.

## Example 1

**Using Green’s theorem, evaluate the closed line integral $\oint_C (x – y) \: dx + (x + y) \: dy$ where $C$ is the positively oriented circle with radius $2$ and centered at the origin. Does Green’s theorem provide a simpler approach to evaluating this line integral?**

We note that all of the conditions for Green’s theorem are satisfied. We have that $P(x, y) = x – y$ and $Q(x, y) = x + y$, so then $\frac{\partial Q}{\partial x} = 1$ and $\frac{\partial P}{\partial y} = -1$ . The region $D$ is the circle with radius $2$ centered at the origin. Let $x = r \cos \theta$ and $y = r \sin \theta$. Then in polar coordinates $D$ can be expressed as:

(2)

Therefore we have that:

(3)

Now let’s evaluate this line integral without using line integrals. We parameterize the circle $C$ as $\vec{r}(t) = (2 \cos t, 2 \sin t)$ for $0 ≤ t ≤ 2\pi$ and so:

(4)

In this case, Green’s theorem is a bit unnecessary due to the how the closed line integral simplifies when evaluated.

## Example 2

**Using Green’s Theorem, evaluate the closed line integral $\oint_C y^3 \: dx – x^3 \: dy$ where $C$ is the positively oriented circle with radius $2$ centered at the origin.**

Once again, we note that all of the conditions for Green’s theorem are satisfied. We have that $P(x, y) = y^3$ and $Q(x, y) = -x^3$. Therefore $\frac{\partial Q}{\partial x} = -3x^2$ and $\frac{\partial P}{\partial y} = 3y^2$.

Furthermore, the region $D$ which is the disk $x^2 + y^2 ≤ 4$ can be best described with polar coordinates once again as:

(5)

Therefore in apply Green’s theorem and using polar coordinates, we have that:

(6)

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