1. **Problem statement:** Verify Green's theorem for the line integral $$\oint_C (2x - y)\,dx + (x + 3y)\,dy$$ where $C$ is the triangle with vertices $(0,0)$, $(1,0)$, and $(0,1)$ traversed counterclockwise. 2. **Green's theorem formula:** $$\oint_C P\,dx + Q\,dy = \iint_D \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dA$$ where $P = 2x - y$ and $Q = x + 3y$. 3. **Calculate partial derivatives:** $$\frac{\partial Q}{\partial x} = \frac{\partial}{\partial x}(x + 3y) = 1$$ $$\frac{\partial P}{\partial y} = \frac{\partial}{\partial y}(2x - y) = -1$$ 4. **Compute the double integral over region $D$ (triangle):** $$\iint_D (1 - (-1)) dA = \iint_D 2 dA = 2 \times \text{area of triangle}$$ The triangle has vertices $(0,0)$, $(1,0)$, $(0,1)$, so its area is $$\frac{1}{2} \times 1 \times 1 = \frac{1}{2}$$ 5. **Evaluate the double integral:** $$2 \times \frac{1}{2} = 1$$ 6. **Calculate the line integral directly:** Parameterize each side of the triangle and compute $$\oint_C (2x - y) dx + (x + 3y) dy$$: - Side 1: $(0,0)$ to $(1,0)$, $y=0$, $x=t$, $t\in[0,1]$ $$dx=dt, dy=0$$ $$\int_0^1 (2t - 0) dt + (t + 0) \times 0 = \int_0^1 2t dt = [t^2]_0^1 = 1$$ - Side 2: $(1,0)$ to $(0,1)$, parameterize as $x=1 - s$, $y=s$, $s\in[0,1]$ $$dx = -ds, dy = ds$$ $$\int_0^1 (2(1 - s) - s)(-ds) + ((1 - s) + 3s)(ds) = \int_0^1 (2 - 2s - s)(-ds) + (1 - s + 3s)(ds)$$ $$= \int_0^1 (2 - 3s)(-ds) + (1 + 2s)(ds) = \int_0^1 (-2 + 3s + 1 + 2s) ds = \int_0^1 (-1 + 5s) ds = [-s + \frac{5}{2}s^2]_0^1 = -1 + \frac{5}{2} = \frac{3}{2}$$ - Side 3: $(0,1)$ to $(0,0)$, $x=0$, $y=1 - u$, $u\in[0,1]$ $$dx=0, dy=-du$$ $$\int_0^1 (2\times0 - (1 - u)) \times 0 + (0 + 3(1 - u))(-du) = \int_0^1 0 - 3(1 - u) du = -\int_0^1 3 - 3u du = -[3u - \frac{3}{2}u^2]_0^1 = -(3 - \frac{3}{2}) = -\frac{3}{2}$$ 7. **Sum of line integrals:** $$1 + \frac{3}{2} - \frac{3}{2} = 1$$ 8. **Conclusion:** Both the double integral and the line integral equal 1, verifying Green's theorem for this problem. --- **Slug:** "greens theorem triangle" **Subject:** "vector calculus" **Desmos:** {"latex":"y=0","features":{"intercepts":true,"extrema":true}} **q_count:** 3