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In this lesson, we will learn how to apply Greenβs theorem to evaluate line integrals as the area integral of the region it bounds in the plane.

Q1:

Let πΆ be the circle with equation π₯ + π¦ = 1 2 2 . Use Greenβs Theorem to evaluate ο 2 π¦ π₯ β 3 π₯ π¦ πΆ d d , where πΆ is traversed counterclockwise.

Q2:

Use Greenβs theorem to evaluate the line integral ο οΉ π₯ β π¦ ο π₯ + 2 π₯ π¦ π¦ πΆ 2 2 d d , where πΆ is the boundary of the region π = ο© ( π₯ , π¦ ) 0 β€ π₯ β€ 1 , 2 π₯ β€ π¦ β€ 2 π₯ ο΅ : 2 , and the curve πΆ is traversed counterclockwise.

Q3:

Use Greenβs theorem to evaluate the line integral ο π₯ π¦ π₯ + 2 π₯ π¦ π¦ πΆ 2 d d , where πΆ is the boundary of π = ο© ( π₯ , π¦ ) 0 β€ π₯ β€ 1 , π₯ β€ π¦ β€ π₯ ο΅ : 2 , traversed counterclockwise.

Q4:

Let πΆ be the boundary of the triangle with vertices ( 0 , 0 ) , ( 4 , 0 ) and ( 0 , 4 ) . Use Greenβs Theorem to evaluate ο οΊ π + π¦ ο π₯ + οΊ π + π₯ ο π¦ πΆ π₯ 2 π¦ 2 2 2 d d , where πΆ is traversed counterclockwise.

Q5:

Evaluate ο οΉ π₯ + π¦ ο π₯ + 2 π₯ π¦ π¦ πΆ 2 2 d d , where πΆ π₯ = π‘ , π¦ = π‘ : c o s s i n and 0 β€ π‘ β€ 2 π .

Q6:

Evaluate ο π₯ π₯ + π¦ π¦ πΆ d d , where πΆ π₯ = 2 π‘ , π¦ = 3 π‘ : c o s s i n and 0 β€ π‘ β€ 2 π .

Q7:

Let πΆ be the boundary of the rectangle with vertices ( 1 , β 1 ) , ( 1 , 1 ) , ( β 1 , 1 ) , and ( β 1 , β 1 ) . Use Greenβs Theorem to evaluate ο π π¦ π₯ + οΉ π¦ + π π¦ ο π¦ πΆ π₯ 3 π₯ s i n d c o s d , where πΆ 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 ?

Q9:

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

Q10:

The figure shows the graph of π ( π₯ ) = β 3 οΌ π₯ + 1 3 ο ( π₯ β 1 ) over the interval [ 0 , 1 ] . Let π be the shaded region and πΆ its boundary, traced counterclockwise. Let F i j ( π₯ , π¦ ) = π¦ + π¦ .

Use Greenβs theorem to calculate ο β ο’ F r d .

Calculate οΈ β ο’ ο F r d , where πΆ ο§ is the line from π to π .

Calculate οΈ β ο’ ο‘ F r d , where πΆ ο¨ is the curve from π to π .

Calculate οΈ β ο’ ο’ F r d , where πΆ ο© is the line from π to π .

Q11:

The figure shows the steps to producing a curve πΆ . 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 πΆ . We let π be the region enclosed by πΆ .

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

Consider the vector field F ( π₯ , π¦ ) = β¨ π¦ , 2 π₯ β© . What is the function π π π₯ β π π π¦ F F 2 1 ?

Use Greenβs theorem to evaluate the line integral οΈ β πΆ F r d , where πΆ is the curve above.

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