Problem: Use Green's Theorem to evaluate the line integral:

$\int_C \log(1+y) \, dx - \frac{xy}{1+y} \, dy$ where $C$ is the boundary of the triangle with vertices $(0, 0)$, $(2, 0)$, and $(0, 4)$.

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Green's Theorem:

Green's theorem states that for a curve $C$ positively oriented and enclosing a region $R$: $\int_C M \, dx + N \, dy = \iint_R \left( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \right) \, dA$ Here:

• $M(x, y) = \log(1 + y)$
• $N(x, y) = -\frac{xy}{1 + y}$

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Step 1: Compute the partial derivatives

1. Compute $\frac{\partial N}{\partial x}$:
   $N(x, y) = -\frac{xy}{1 + y}, \quad \frac{\partial N}{\partial x} = -\frac{y}{1 + y}$

2. Compute $\frac{\partial M}{\partial y}$:
   $M(x, y) = \log(1 + y), \quad \frac{\partial M}{\partial y} = \frac{1}{1 + y}$

3. Compute $\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y}$:
   $\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} = -\frac{y}{1 + y} - \frac{1}{1 + y} = -\frac{y + 1}{1 + y} = -1$

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Step 2: Set up the double integral over $R$

The region $R$ is the triangle with vertices $(0, 0)$, $(2, 0)$, and $(0, 4)$.
The limits of integration can be described as:

• $x$ ranges from $0$ to $2$
• For a fixed $x$, $y$ ranges from $0$ to $4 - 2x$

Thus: $\iint_R -1 \, dA = -\iint_R 1 \, dA = -\int_0^2 \int_0^{4 - 2x} 1 \, dy \, dx$

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Step 3: Evaluate the double integral

1. Compute the inner integral ($y$-integration):
   $\int_0^{4 - 2x} 1 \, dy = [y]_0^{4 - 2x} = (4 - 2x) - 0 = 4 - 2x$

2. Compute the outer integral ($x$-integration):
   $\int_0^2 (4 - 2x) \, dx = \int_0^2 4 \, dx - \int_0^2 2x \, dx$ $\int_0^2 4 \, dx = 4x \Big|_0^2 = 8, \quad \int_0^2 2x \, dx = x^2 \Big|_0^2 = 4$ $\int_0^2 (4 - 2x) \, dx = 8 - 4 = 4$

3. Multiply by $-1$:
   $-\iint_R 1 \, dA = -4$

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Final Answer:

$\int_C \log(1+y) \, dx - \frac{xy}{1+y} \, dy = -4$

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Do you have any questions or need clarification? 😊
Here are 5 related problems to explore:

1. Apply Green’s theorem to evaluate a line integral where $M$ and $N$ are more complex functions.
2. Determine the area of a triangle using Green’s theorem.
3. Solve a similar problem where $C$ is a rectangle instead of a triangle.
4. Verify Green's theorem numerically for a given function.
5. Explore the conditions under which Green's theorem can be applied.

Tip: Always ensure the curve $C$ is oriented positively (counterclockwise) when using Green’s theorem!