 Now, let's look at what happens when we have the work for variable forces. As a reminder, work is the transfer of energy from a force causing a displacement. Our basic equation for constant force was our work is the f delta r cosine theta. And it assumes several things. It assumes that the magnitude of the force is constant, the direction of the force is constant, and the displacement is in a straight line. If I've got a changing force, then I need to adjust my process. Now, hopefully, I'll be able to divide the entire path up into segments, where on each of those segments, the force is constant along the segment. If that's true, then I can calculate the work for each of the segments and then add up the total work to find the full energy transfer. I want to start with a little bit of a simpler case where the force is all in the x direction. I've got no y force at all. But I've got a change in the magnitude of the force at different positions along that motion. Well, if this is the case where I've got a force in the x direction, that means that I always have a cosine of 0 degrees, which is 1. And so I can find my work a little bit more simply. So for example, for my first segment, my delta x is the width of that segment. The force 1 is the height of that segment. And the work is the force times the displacement. Again, I have a cosine of 0 degrees, which becomes 1. Well, this force times the displacement is also the area of this rectangle. So the work done during the first segment is the area of the rectangle on the curve. I can use a similar approach to find the work done in the second segment, which is also a rectangle. But now I've got a new delta x and f. My third segment is still a rectangle and my fourth section is still a rectangle. And if I was given exact values on this graph for what the force and the positions were, I could calculate out all four of those quantities and add them up to give me the net work. Here's a second one, where instead of having rectangles, I actually have a force which is changing over the particular segment. Well, I still have this concept, though, that that work is related back to the area of the shape. In this case, my first segment is not a rectangle, but a triangle. But the area of a triangle is 1 half the base times the height. In this particular case, that's 1 half, now my base is my delta x, my height is my f. But I could also think of this 1 half f as being the average force over that particular displacement. So I've found the force and the displacement for this segment, I can find the work for that segment. Now in this case, my second one is a rectangle, and so I can use just the force times the displacement. My third segment is again a triangle, so I've got the 1 half the force times the displacement. And again, this force is the highest force or the height of that particular triangle. And once again, it adds up to give me my net force if I add up all three of those areas. I'm going to find the total work done as I move across that displacement. So in general, the work done is the area under the force and position plot. If it's nice geometric sections, it's easy to find the area of each segment. It might be a rectangle, it might be a triangle, but we can find all of those areas. And then we add up the total amount of work done, all of the area of all those geometric sections. Now if my force versus position plot is something that's curved, I'm going to need calculus to find the area under that curve. And I can use calculus to find my work. But in general, I can use the area. So that wraps up our variable forces. Again, this is just an introduction to the concept of it. We'll look at some more examples later.