Use Green's Theorem to evaluate F. dr. (Check the orientation of the curve before applying the theorem.) F(x, y) = (y? cos(x), x² + 2y sin(x)) C is the triangle from (0, 0) to (1, 3) to (1, 0) to (0, 0)
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Integration
Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.
Application of Integration
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![### Application of Green's Theorem
To evaluate the line integral \(\oint_C \mathbf{F} \cdot d\mathbf{r} \) using Green's Theorem, follow these instructions. It is important to check the orientation of the curve \(C\) before applying the theorem.
Given:
\[
\mathbf{F}(x, y) = \langle y^2 \cos(x), x^2 + 2y \sin(x) \rangle
\]
Curve \(C\) forms a triangle with vertices at:
1. \( (0, 0) \)
2. \( (1, 3) \)
3. \( (1, 0) \)
4. Returning to \( (0, 0) \)
These vertices describe a triangle in the coordinate plane. Ensure the curve is oriented counterclockwise to apply Green’s Theorem correctly.
**Steps to Apply Green’s Theorem:**
1. **Vector Field Components**: Identify \( \mathbf{F} = (P, Q) \), where
\[
P(x, y) = y^2 \cos(x)
\]
\[
Q(x, y) = x^2 + 2y \sin(x)
\]
2. **Green’s Theorem Formula**:
\[
\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_D \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dA
\]
where \(D\) is the region enclosed by \(C\).
3. **Calculate Partial Derivatives**:
\[
\frac{\partial Q}{\partial x} = \frac{\partial}{\partial x} (x^2 + 2y \sin(x)) = 2x + 2y \cos(x)
\]
\[
\frac{\partial P}{\partial y} = \frac{\partial}{\partial y} (y^2 \cos(x)) = 2y \cos(x)
\]
4. **Integrand Simplification**:
\[
\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = (2x + 2y \cos(x))](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc6615292-9c23-445f-b631-819ab1904cf2%2F49545313-d53a-49a4-a98f-c6efa9480d13%2Fyzguyxi.png&w=3840&q=75)
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