section 7.5 the work done by a general variable force so the last section we saw how we could calculate the work done by a variable force the spring force so that was very useful for if we don't have a constant force if we have a force that matches that form of hooke's law negative kx we can integrate it now what if we have a more general variable force what do we do then well look at a one dimensional example still thinking of the x dimension and just as we saw with the spring force we're going to need to integrate the work equation the one that normally only applies for the constant force case over the change in position so if we have uh we can illustrate this integration if we have a graph of force versus position right and if the force is varying over time then we can fill in okay the force is roughly constant for a narrow interval right and over that interval where the force is constant we can multiply by the displacement of that little sliver right that's what's shown in these big ones there's an average value of the work and so you can get a little tiny work segment and then we can just add up all those little work segments we can keep shrinking down the segments the delta x's until it becomes infinitesimally small right and we end up with just an integration so we look at where the strip widths go down and the limit of them approaching zero where we're just integrating each little tiny piece and so ultimately to get the work we just want the area under the curve under the force curve as a function of displacement or position so what does this mean for the math are some of the rectangles we could write is the sum of the average force for each little sliver times the width of that we're going to look at the limit where the width of each of those rectangles goes to zero which just becomes an integral right integral become of the force times dx from some initial to some final position and in three dimensions we can integrate each one separately which is pretty nifty the other useful thing here is that the work kinetic energy theorem still applies so if you are able to integrate the force over position you can set that work that you calculated equal to the change in kinetic energy so if you know how the force depends on x you could do that integration right you could integrate it as we showed with the spring force you could even do a graphical integration if you had a variable force as shown sometimes it might be a more linear and you could actually just calculate the area under the curve and know that that is the work once you know the work you know the change in kinetic energy so there's a lot that you can learn even if it's a variable force which is harder to work with so we won't do this quite as often as working with our constant forces but it's useful to be able to see this as another method another tool to add to our arsenal of approaches for solving physics problems