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Work in a variable force field is calculated by integrating the force with respect to displacement.

When a force is constant, work can be calculated by multiplying the force by the displacement and the cosine of the angle between them. However, in a variable force field, the force changes as the object moves, so this method cannot be used. Instead, we must use calculus to integrate the force with respect to displacement.

The formula for work in a variable force field is:

W = ∫ F(x) dx

where W is the work done, F(x) is the force at a given position x, and dx is the displacement. This integral represents the area under the force-displacement curve.

To calculate the work done by a variable force, we must first determine the force function F(x). This can be done using physical laws or experimental data. Once we have the force function, we can integrate it with respect to displacement to find the work done.

For example, if the force function is F(x) = 2x + 3, and the object moves from x = 1 to x = 5, the work done is:

W = ∫ F(x) dx from x=1 to x=5

W = ∫ (2x + 3) dx from x=1 to x=5

W = [x^2 + 3x] from x=1 to x=5

W = (5^2 + 3(5)) - (1^2 + 3(1))

W = 25 + 15 - 1 - 3

W = 36

Therefore, the work done by the variable force in this example is 36 units.

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