 Hello, and welcome to this quick recap on section 6.1, using definite integrals to find area and length. In this section, we're going to use the definite integral in some new ways that are still closely related to the way we first defined it. So first, we'll recap a definite integral. The definite integral of a function f of x from a to b represents the net signed area between the graph of f of x and the x-axis on that interval. The way we defined this initially was by making some approximations. On the left, these are some boxes, in this case using a left-hand rule, that approximate that area. In the middle, as we added more and more boxes, we got a better and better approximation. Until finally, on the right, we get the exact area under the curve with the area above the x-axis being positive and the area below the x-axis being negative. In this section, we're going to use a similar approach with some slightly different contexts. We're going to identify some quantity in which we're interested, such as the area between two curves. We're going to write it as a sum of many tiny parts, just like we wrote tiny arbitrarily thin boxes to define the definite integral. And we're going to rewrite this as a definite integral. The big picture idea here is that a definite integral adds up an arbitrary number of very thin boxes or other objects and gives you an exact value for them. Let's take a look at the first example. Suppose we have two functions, and instead of finding the area under just one of them, we want to find the area between them. By the area between them, we mean between the points where they intersect. And on this picture, that means we want the area from here all the way up to there. Well, if we drew a picture of the boxes under just curve g, the red one, that would be something like this. Now we'll draw the boxes under just curve f. And we can see that they take up some, but not all, of the area under curve g. If we subtracted away these blue boxes, we would be left with this, which is just the area between the two curves. And that's exactly what we're going to do in this part. We typically only draw one box, and that box goes from the top curve down to the bottom curve. The width of this box is arbitrarily thin in the x direction. We'll call it delta x, exactly the same as when we defined an integral. Delta x is how wide the box is. The height of the box is a little different here. It goes from the g curve down to the f curve, and so its height is g of x minus f of x. We call this box a representative rectangle, and we usually only draw it so that it's easier to tell what we're doing on any given picture. So the result we have here says if we have two curves, g of x and f of x, and they intersect at a and b, and g is always above f on the interval a to b, then the area between the curves is given by this integral where we take g of x and subtract f of x, and the integral goes from a to b. The g minus f part of this comes from the height of the box that we drew in right here. We can use this in another situation as well. It's possible for us to plot functions on the side, so to speak, which is to say we can plot a function of y instead of a function of x. You may want to review some algebra to remind yourself how to plot a function of y. In this case, we have two curves that are functions of y, and we see that if we tried to draw vertical boxes like we did before, that they would sometimes go from the red curve down to the red curve, and sometimes they'd go from the red curve down to the blue curve, giving us an inconsistent formula for which curve is on top and which is on the bottom. If instead we put a horizontal box in here, as in the right picture, this always goes from the blue curve over to the red curve. The height of this box is delta y, a very thin slice in the y direction, and its width is always going from the blue curve down to the red curve, just like before we had a height from the top curve down to the bottom curve. So we end up with a very similar result. If you have two functions of y, and they intersect at C and D, and the function g is always bigger than function y. Remember, for functions of y, this means g is farther to the right. Then the area between the curves is this integral, where we take g and subtract f from it, and we integrate from C to D. Again, we've drawn just a representative rectangle on the right to show what the height and width of the box would be in general. Finally, our last use of definite integrals in this section is to find the length of a curve, also called arc length. In this case, the length of a curve is how far you would walk if you started at one point, such as right here, and walked along the curve to another endpoint, such as right here. We can approximate this using straight line segments, like shown in green here. And as you can read in the book, each of these straight line segments is the hypotenuse of a right triangle, and it has a width delta x and a height delta y in this case. Once we see those tiny slices in the x and y direction, that's a hint that we're going to add up a bunch of very thin things using an integral. And so the result here is that the total arc length along a differentiable function, f of x, from x equals a to x equals b, is given by this integral, where we integrate the square root of one plus the derivative of x squared. Notice that the whole derivative is squared there, and we integrate from a to b. Now that we've seen these functions, let's take a look at how to use them in some examples.