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The image contains a set of calculus exercises, specifically focusing on line and surface integrals, vector fields, Green's Theorem, and other related topics. Here’s a breakdown of the exercises:

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### 1. **Calculate the line integral**:  
   \[
   \int_C \left[ (x + y)dx + (y - x)dy \right]
   \]
   along the curves:  
   (a) the parabola \( x = y^2 \)  
   (b) a straight line  
   (c) another straight line from (1,1) to (1,2)  
   (d) a parametric curve \( x = 2t^2 + t + 1, y = t^2 + 1 \)

### 2. **Calculate the line integral**:  
   \[
   \int_C \left[ (2x - y + 4)dx + (5y + 3x - 6)dy \right]
   \]
   around the triangle defined by points (0,0), (3,0), and (3,2), counterclockwise.

### 3. **Given the vector field** \( \mathbf{F} = (x^2 - y^2)\hat{i} + 2xy\hat{j} \), calculate  
   \[
   \oint_C \mathbf{F} \cdot d\mathbf{r}
   \]
   over a curve described by \( y = x(x-1) \) between points (1,0) and (2,2).

### 4. **Use Green's Theorem** to compute the line integral  
   \[
   \oint_C \mathbf{F} \cdot d\mathbf{r}
   \]
   where \( \mathbf{F} \) varies for three different cases, and the curve encloses different regions. For example:  
   (a) \( \mathbf{F} = 3x^2\hat{i} - 4xy\hat{j} \), for a rectangular region  
   (b) \( \mathbf{F} = y\hat{i} - x\hat{j} \), for a circular region  
   (c) \( \mathbf{F} = (x^3 - 3y)\hat{i} + (x + \sin y)\hat{j} \), for a triangular region.

### 5. **Verify Green's Theorem** for the field  
   \[
   \mathbf{F} = (x^3 - y^2)\hat{i} + (x^2 y)\hat{j}
   \]
   in the region bounded by the circle \( x^2 + y^2 = 4 \).

### 6. **Surface Integral**:  
   Compute the surface integral  
   \[
   \int_S (x + z^2) \, dS
   \]
   where \( S \) is the surface defined by \( z^2 = 3(x^2 + y^2) \).

### 7. **Calculate the area of the region** formed by the intersection of two cylinders, defined by  
   \[
   x^2 + y^2 = 9 \quad \text{and} \quad z^2 + y^2 = 9.
   \]

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### Would you like me to help solve any specific question from the list?

Here are five relative questions:
1. How can Green's Theorem simplify the calculation of line integrals over closed curves?
2. What is the physical interpretation of a surface integral?
3. How do we parameterize curves for line integrals in complex forms like parametric curves?
4. How can symmetry help simplify the surface integral over a cylinder?
5. How does the choice of vector field influence the result of a line integral?

**Tip**: Green's Theorem can significantly reduce the complexity of line integrals when working with closed curves by converting them to double integrals over the enclosed region.

The image contains several calculus exercises focusing on line and surface integrals, Green's Theorem, and vector fields.

Explore a set of calculus problems focusing on line and surface integrals, and Green's Theorem. Learn to apply these concepts to vector fields and regions bounded by curves like parabolas and triangles.

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