Use Green's Theorem to evaluate 
 
F · dr.

C
 (Check the orientation of the curve before applying the theorem.)
F(x, y) = 

y cos x − xy sin x, xy + x cos x

, C is the triangle from (0, 0) to (0, 10) to (2, 0) to (0, 0)

To evaluate \( \oint_C \mathbf{F} \cdot d\mathbf{r} \) using Green's Theorem, we need to apply the theorem, which states:

\[
\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_D \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \, dA
\]

where \( \mathbf{F}(x, y) = (P(x, y), Q(x, y)) \) and \( D \) is the region enclosed by \( C \). In our case:

\[
P(x, y) = y \cos x - xy \sin x
\]
\[
Q(x, y) = xy + x \cos x
\]

### Step 1: Calculate \( \frac{\partial Q}{\partial x} \) and \( \frac{\partial P}{\partial y} \)

1. **Calculate \( \frac{\partial Q}{\partial x} \):**

   \[
   \frac{\partial Q}{\partial x} = \frac{\partial}{\partial x} (xy + x \cos x) = y + \cos x - x \sin x
   \]

2. **Calculate \( \frac{\partial P}{\partial y} \):**

   \[
   \frac{\partial P}{\partial y} = \frac{\partial}{\partial y} (y \cos x - xy \sin x) = \cos x - x \sin x
   \]

### Step 2: Substitute into the Green's Theorem formula

Now we substitute into the Green's Theorem formula:

\[
\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_D \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \, dA
\]

Substituting \( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \):

\[
\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = \left( y + \cos x - x \sin x \right) - \left( \cos x - x \sin x \right) = y
\]

So we get:

\[
\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_D y \, dA
\]

### Step 3: Set up the double integral over the triangular region

The region \( D \) is the triangle with vertices at \( (0, 0) \), \( (0, 10) \), and \( (2, 0) \). To set up the double integral, we can integrate \( y \) with respect to \( x \) and \( y \) over this region.

In this triangle, \( x \) ranges from 0 to 2, and for a fixed \( x \), \( y \) ranges from 0 to \( 10 - 5x \) (the equation of the line from \( (0, 10) \) to \( (2, 0) \)).

\[
\iint_D y \, dA = \int_0^2 \int_0^{10 - 5x} y \, dy \, dx
\]

### Step 4: Evaluate the integral

1. Integrate with respect to \( y \):

   \[
   \int_0^2 \int_0^{10 - 5x} y \, dy \, dx = \int_0^2 \left[ \frac{y^2}{2} \right]_0^{10 - 5x} \, dx
   \]

   \[
   = \int_0^2 \frac{(10 - 5x)^2}{2} \, dx
   \]

2. Simplify and integrate with respect to \( x \):

   Expanding \( (10 - 5x)^2 \):

   \[
   = \int_0^2 \frac{100 - 100x + 25x^2}{2} \, dx
   \]

   \[
   = \int_0^2 \left( 50 - 50x + \frac{25x^2}{2} \right) \, dx
   \]

3. Now integrate each term:

   \[
   = \left[ 50x - 25x^2 + \frac{25x^3}{6} \right]_0^2
   \]

   Substituting \( x = 2 \):

   \[
   = 50(2) - 25(2)^2 + \frac{25(2)^3}{6}
   \]

   \[
   = 100 - 100 + \frac{200}{6} = \frac{200}{6} = \frac{100}{3}
   \]

### Final Answer:

\[
\oint_C \mathbf{F} \cdot d\mathbf{r} = \frac{100}{3}
\]

Would you like further clarification on any part of this solution?

---

**Additional Related Questions:**

1. How does Green’s Theorem relate to the Fundamental Theorem of Calculus?
2. What happens if the orientation of the curve \( C \) is reversed?
3. Can Green’s Theorem be applied to regions with holes?
4. How would the result change if \( \mathbf{F} \) had a different orientation?
5. How can Green’s Theorem be extended to three-dimensional regions?

**Tip:** When using Green's Theorem, ensure the curve's orientation is counterclockwise to match the positive orientation for the theorem's application.

Use Green's Theorem to evaluate F · dr. C (Check the orientation of the curve before applying the theorem.) F(x, y) = y cos x − xy sin x, xy + x cos x, C is the triangle from (0, 0) to (0, 10) to (2, 0) to (0, 0)

Learn how to evaluate the line integral of F(x, y) = (y cos x − xy sin x, xy + x cos x) using Green's Theorem over the triangular region from (0, 0) to (0, 10) to (2, 0). This solution covers key concepts of multivariable calculus, including line integrals, Green's Theorem, and double integrals. Suitable for undergraduate calculus students.

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