# Worksheet: Green's Theorem

In this worksheet, we will practice applying Greenβs theorem to evaluate line integrals as the area integral of the region it bounds in the plane.

**Q1: **

Let be the circle with equation . Use Greenβs Theorem to evaluate , where is traversed counterclockwise.

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**Q2: **

Use Greenβs theorem to evaluate the line integral , where is the boundary of the region , and the curve is traversed counterclockwise.

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**Q3: **

Use Greenβs theorem to evaluate the line integral , where is the boundary of , traversed counterclockwise.

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**Q4: **

Let be the boundary of the triangle with vertices and . Use Greenβs Theorem to evaluate , where is traversed counterclockwise.

**Q5: **

Evaluate , where and .

**Q6: **

Evaluate , where and .

**Q7: **

Let be the boundary of the rectangle with vertices , and . Use Greenβs Theorem to evaluate , where is traversed counterclockwise.

**Q8: **

Use Greenβs theorem to determine the conditions on , , , and for the vector field to be conservative. In that case, what is the potential function for that satisfies ?

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**Q9: **

Use Greenβs theorem to compute the line , where is the circle of radius and center at the origin and .

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**Q10: **

The figure shows the graph of over the interval . Let be the shaded region and its boundary, traced counterclockwise. Let .

Use Greenβs theorem to calculate .

Calculate , where is the line from to .

Calculate , where is the curve from to .

Calculate , where is the line from to .

**Q11: **

The figure shows the steps to producing a curve . It starts as the boundary of the unit square in Figure (a). In Figure (b), we remove a square quarter of the area of the square in (a). In Figure (c), we add a square quarter of the area that we removed in (b). In Figure (d), we remove a square quarter of the area of the square we added in (c). If we continue to do this indefinitely, we will get the curve . We let be the region enclosed by .

By summing a suitable series, find the area of region . Give your answer as a fraction.

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Consider the vector field . What is the function ?

Use Greenβs theorem to evaluate the line integral , where is the curve above.

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