 Now let's take a look at the work done by variable forces using a calculus-based approach. Well, as a reminder, work was the transfer of energy due to a force causing a displacement. And if I had a constant force, it was given by this equation, where I had the dot product between the force vector and the displacement vector. But again, that assumes I have a single value for the force and displacement and that they're always in the same directions relative to each other. If that's not true, I can't use this equation anymore. So what do I do? Well, I introduce the concept of incremental paths. And by this, I mean, if I can divide that displacement path into smaller regions where the force is approximately constant along each little section of displacement, then I could define my work as the total of the dot product between the force and displacement for each little segment. Segment 1, segment 2, segment 3, however many segments I need. Well, mathematically, when we're adding up individual terms like this, we can write this as the summation over the dot product between the force and displacement for individual little sections. And as I reduce the size of those sections, the summation becomes the integral where my delta r becomes just the derivative of r. So this is the integral of the force with respect to the displacement. And we need to be a little careful there because that integral had a dot product. It wasn't just force and displacement. It was the force dot product with the dr vector. Well, remember that if I've got the dot product, I can break that down into components, which means I can do the fx dx, the fydy, and if I've got three dimensions, an fcdz, all separately and then add up the results. Well, since work is not a vector, I can add up the three integrals as well. It's also important to keep in mind that I've got limits of my integration. So I need a starting and ending point for this path. So in general, I could say that that's my initial position and my final position, my ri, my rf. If I'm going to break it down into components, that means each one of these integrals has an initial and final position for that particular component. Now, if I only have a force in the x direction that has no y and z component for the force, then I can ignore these last two terms and do only the integral of f of x dx. Similarly, if I had something that was only y, I could ignore the x and the z integrals. But in general, you want to make sure that you're taking into account all three directions and just keep track of which one of these really matters for an individual problem. So that's the calc-based equation for variable forces. We'll see specific examples for individual variable forces, such as the spring force in a later video.