Evaluate the line integral $\oint_C [(y^2 + 2x) dx + (3xy + 4y) dy]$ counter-clockwise along the boundary $C$ of the square bounded by $x = 0$, $x = 2$, $y = 0$, and $y = 2$. (Give the exact integer value)

Answer Key:

4

### 5-Stage Derivation & Analytical Solution: 1. **Stage 1: Green's Theorem:** $\oint_C (P dx + Q dy) = \iint_R \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) dx dy$. 2. **Stage 2: Partial Derivatives:** * $P(x,y) = y^2 + 2x \implies \frac{\partial P}{\partial y} = 2y$. * $Q(x,y) = 3xy + 4y \implies \frac{\partial Q}{\partial x} = 3y$. * Curl term: $\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 3y - 2y = y$. 3. **Stage 3: Integration Over Square Region $[0,2] \times [0,2]$:** * $I = \int_{x=0}^2 \int_{y=0}^2 y \, dy \, dx = \left(\int_{0}^2 dx\right) \left(\int_0^2 y \, dy\right)$. * $\int_0^2 dx = 2$. * $\int_0^2 y \, dy = \left[\frac{y^2}{2}\right]_0^2 = \frac{4}{2} = 2$. * $I = 2 \times 2 = 4$. 4. **Stage 4: Common Trap:** Forgetting to subtract $\partial P/\partial y$ from $\partial Q/\partial x$ or sign error on circulation. 5. **Stage 5: Official Reference:** Erwin Kreyszig, *Advanced Engineering Mathematics*, Chapter 10.

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