Green's Theorem states that for a positively oriented, piecewise smooth, simple closed curve $C$ enclosing a region $R$:

$\oint_C \left( P \, dx + Q \, dy \right) = \iint_R \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dA$

Here, we have:

• $P(x, y) = x^2 + y^4$
• $Q(x, y) = x^2 - y^2$

The region $R$ is the triangle with vertices $(0, 0)$, $(0, 3)$, and $(9, 3)$.

---

Step 1: Compute partial derivatives

$\frac{\partial Q}{\partial x} = \frac{\partial}{\partial x}(x^2 - y^2) = 2x$ $\frac{\partial P}{\partial y} = \frac{\partial}{\partial y}(x^2 + y^4) = 4y^3$

The integrand for the double integral becomes: $\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 2x - 4y^3$

---

Step 2: Set up the limits of integration

The triangle is bounded by:

• $y = 0$ (base),
• $y = 3$ (top),
• $x = 0$ to $x = 3y$ (right slanted edge).

Thus, the region $R$ can be described with: $0 \leq y \leq 3, \quad 0 \leq x \leq 3y$

---

Step 3: Evaluate the double integral

$\iint_R \left( 2x - 4y^3 \right) dA = \int_0^3 \int_0^{3y} \left( 2x - 4y^3 \right) dx \, dy$

Integral with respect to $x$:

$\int_0^{3y} \left( 2x - 4y^3 \right) dx = \left[ x^2 - 4y^3x \right]_0^{3y}$ $= \left( (3y)^2 - 4y^3(3y) \right) - \left( 0^2 - 4y^3(0) \right)$ $= 9y^2 - 12y^4$

Integral with respect to $y$:

$\int_0^3 \left( 9y^2 - 12y^4 \right) dy = \int_0^3 9y^2 \, dy - \int_0^3 12y^4 \, dy$

Evaluate each term:

1. $\int_0^3 9y^2 \, dy = 9 \int_0^3 y^2 \, dy = 9 \left[ \frac{y^3}{3} \right]_0^3 = 9 \left( \frac{27}{3} \right) = 81$
2. $\int_0^3 12y^4 \, dy = 12 \int_0^3 y^4 \, dy = 12 \left[ \frac{y^5}{5} \right]_0^3 = 12 \left( \frac{243}{5} \right) = \frac{2916}{5}$

Combine the results: $\int_0^3 \left( 9y^2 - 12y^4 \right) dy = 81 - \frac{2916}{5} = \frac{405}{5} - \frac{2916}{5} = -\frac{2511}{5}$

---

Final Answer:

$\oint_C \left( (x^2 + y^4) \, dx + (x^2 - y^2) \, dy \right) = -\frac{2511}{5}$

---

Would you like further details on the steps or a graph of the region?

1. What is Green's Theorem?
2. How do partial derivatives relate to the theorem?
3. Why do we need limits of integration?
4. Can this be solved without Green's Theorem?
5. How to visualize such regions geometrically?

Tip: Always check the orientation of the curve before applying Green’s Theorem.