Question

Use Green's theorem to calculate the work done by the force F on a particle that is moving counterclockwise around the closed path C.

Answer 1

Step 1:

The vector field is and vertices of the triangle are

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Greens theorem :

If C  be a positively oriented closed curve, and R be the region bounded by C, M and N are

the partial derivatives on an open region then.

Graph :

(1) Draw the coordinate plane.

(2) Plot the vertices .

(3) Connect the plotted vertices to a smooth triangle.

Observe the graph :

The limits of y  are varying from 0 to 5, so .

Find the bounds for x :

Lower limit :

Consider the points .

From the points, coordinates are equal then the equation of the line parallel to axis.

So the equation of the line is .

Lower limit of x is .

Upper limit :

Consider the points .

Using two points form of a line equation is .

Substitute in the line equation.

Upper limit of x is .

Therefore the limits of x is to , so .

Step 2:

Using greens theorem,

The region bounded by the triangle is .

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Solution :

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