Worksheet: Green's Theorem

Practice

In this worksheet, we will practice applying Green’s 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 Green’s Theorem to evaluate , where is traversed counterclockwise.

Q2:

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

Q3:

Use Green’s theorem to evaluate the line integral , where is the boundary of , traversed counterclockwise.

Q4:

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

Q8:

Use Green’s 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 ?

Q9:

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

Q10:

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

Use Green’s 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.

Consider the vector field . What is the function ?

Use Green’s theorem to evaluate the line integral , where is the curve above.