How Do You Calculate Work Done by a Variable Force Along a Displacement?

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To calculate the work done by a variable force, the equation Fx = (2x + 4) N is used for a 5.00 kg particle moving from x = 2.60 m to x = 6.20 m. The work done is determined through integration of the force over the displacement, specifically from x = 2.6 to x = 6.2. While the force is linear, the average force can be calculated using the values at the endpoints, but integration provides the exact work done. The correct approach involves integrating the force function rather than simply applying the work formula W = FD. Therefore, the work done must be calculated through the integral of the force function across the specified displacement.
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A single conservative force acts on a 5.00 kg particle. The equation Fx = (2x + 4) N describes this force, where x is in meters. As the particle moves along the x-axis from x = 2.60 m to x = 6.20 m, calculate the following.
(a) the work done by this force
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Alright this seems pretty straight forward...W=FD, distance equals the change in X which is (6.2-2.6) which equals 3.6...The force acting on the particle is where I get a bit confused. F=2x+4...do you do it with x=6.2 or 2.6? Or do you find the value of 6.2 and subtract is from 2.6? Thanks for any help
 
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In the general case, you would do an integration.

WD = int, from x=2.6 to x=6.2, of Fx dx

However, in this particular case, since Fx is linear, then you can just take the average Force.

F_av = (F_6.2 + F_2.6)/2.
 
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The force is not constant. You need to integrate to find the work done.
 

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