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$Use Green's Theorem to evaluate the following line integral. Assume the curve is oriented counterclockwise The circulation line integral of F=(:3xy^(2),3x^(3)+y:) where C is the boundary of {(x,y):0 <= y <= sin x,0 <= x <= pi}$

Use Green's Theorem to evaluate the following line integral. Assume the curve is oriented counterclockwise $\newline$ The circulation line integral of $F=\left\langle 3 x y^{2}, 3 x^{3}+y\right\rangle$ where $C$ is the boundary of $\{(x, y): 0 \leq y \leq \sin x, 0 \leq x \leq \pi\}$

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Q. Use Green's Theorem to evaluate the following line integral. Assume the curve is oriented counterclockwise

\newline

The circulation line integral of

F=\left\langle 3 x y^{2}, 3 x^{3}+y\right\rangle

where

C

is the boundary of

\{(x, y): 0 \leq y \leq \sin x, 0 \leq x \leq \pi\}

Identify Vector Field and Region: Identify the vector field and the region over which the integral is to be evaluated. $F = (3xy^2, 3x^3 + y)$ Region R: $0 \leq y \leq \sin(x)$ , $0 \leq x \leq \pi$
Write Green's Theorem: Write down Green's Theorem. $\newline$ Green's Theorem relates a line integral around a simple, closed, positively oriented curve $C$ to a double integral over the region $D$ enclosed by $C$ : $\newline$ $\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_D \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) dA$ $\newline$ where $\mathbf{F} = (P, Q)$
Calculate Partial Derivatives: Calculate partial derivatives needed for Green's Theorem. $\newline$ $P = 3xy^2$ , $Q = 3x^3 + y$ $\newline$ $\frac{dP}{dy} = 6xy$ , $\frac{dQ}{dx} = 9x^2$
Set Up Double Integral: Set up the double integral using the partial derivatives. $\iint_D (9x^2 - 6xy) \, dA$
Express Limits in Terms of x and y: Express the double integral in terms of x and y limits. $\newline$ $\int_{x=0}^{x=\pi} \int_{y=0}^{y=\sin(x)} (9x^2 - 6xy) \, dy \, dx$
Integrate with Respect to y: Integrate with respect to y first. $\newline$ $\int_{y=0}^{y=\sin(x)} (9x^2 - 6xy) dy = [9x^2y - 3xy^2]_{y=0}^{y=\sin(x)}$ $\newline$ $= 9x^2\sin(x) - 3x\sin^2(x)$
Integrate with Respect to x: Integrate with respect to $x$ . $\newline$ $\int_{x=0}^{x=\pi} (9x^2\sin(x) - 3x\sin^2(x)) \, dx$ $\newline$ This integral needs to be evaluated, typically using integration techniques or numerical methods.