 Hi everyone and welcome to this lesson on finding the area between two curves. So let's start with a very basic geometry problem. Suppose I had a circle inscribed inside of a square and I wanted to find those blue corners and the area of them in total. So if we had some dimensions we could find the area of the square and subtract from that the area of the circle. So hopefully that's what you thought to do. So let's apply this idea now to finding area between two curves. So let's build up to that idea. Suppose I just had a curve such as this one and I wanted to find the area under this curve between A and B. So hopefully what you're thinking of is that we could write an integral to do that and we could do the integral from A to B of f of x dx and that would enable us to find that exact area. So suppose we had another curve and same question. Now I want the area under g of x in between A and B. Once again I could write an integral to help me find that area. So let's now combine the two. So now I want to find the area that's in between f and g. Any thoughts on how we could do that? Think about our circle that was inscribed inside the square. Well if all I'm concerned with is this area in between here. If I could find the length, if you want to think of it that way, from the f of x curve on the top to the g of x curve on the bottom that would allow me to find the area by doing an integral. So I could do the integral from A to B of f of x minus g of x dx. So let's talk about the theory a little bit behind this. So if we have two functions f and g, both of which are continuous on the closed interval from A to B and we know that one of the functions sits higher than the other for all x's on that interval from A to B and we wish to find the area that's bounded by those two curves in between the x values of A and B. The idea goes back to what you learned about Riemann's sums. So let's go back to our picture. With Riemann's sums you learned how you could insert into an area rectangles the sum of whose areas would approximate the area under the curve. Well imagine doing that for this. Imagine using rectangles. Now to make things easy we typically do use the rectangles of the same width but obviously the height's going to differ. So imagine having lots and lots of rectangles inscribed in this area. You could find the area of each individual rectangle then add them up. That's really what you're doing here. The height of each individual rectangle is given by f of x minus g of x. The width is your delta x just like you had learned with Riemann's sums. So each width of the rectangle is delta x. The height is f of x minus g of x. So you could think of it in terms of a summation just like you learned prior to learning definite integrals. So the limit as n approaches infinity remember that means as you have more and more rectangles inscribed in that region and you're doing the summation of your heights that would be the f of x minus g of x times the width of each rectangle and adding all of those up. And that's really going to be the same as doing a definite integral from a to b of f of x dx. So let's talk about the general steps then for how to find the area. The first thing you'll want to do is graph the curves on your graph and calculator and find the limits of integration. Sometimes you'll be told what those limits are. Sometimes you might have to find the points of intersection of the curves yourself. If you do, and especially if they're longer decimal values you will want to store those found values in your calculator. Then you'll have to consider either a vertical or horizontal rectangle in that desired region. I like to think of it as a representative rectangle. Go back to the previous picture if you would. In this case we had vertical representative rectangles. Sometimes it's helpful to think of the orientation of what one single rectangle would look like when you go to do a problem like this. So once you've decided if it's going to be a vertical or horizontal rectangle your guide for deciding that should be that you want the rectangle to touch both curves if at all possible. Because that leads into you're then knowing whether it's going to be a dx or a dy problem. If you have a vertical rectangle your widths are dx's. They're delta x's. That's because the width lies horizontally. Your height then as we saw in that previous picture is f of x minus g of x essentially the y value at the top minus the y value at the bottom. Your limits of integration are x values. One thing to take note if it's dx your limits of integration are x's. Now once in a while you'll run into a problem where you have a horizontal oriented rectangle that's going left to right and maybe it's hitting one curve on the left another curve on the right. In that case you have a dy problem. The reason is the width is going up and down so it's a delta y this time. The width or if you want to think length of your rectangle then is the x value on the right minus the x value on the left. And again the important thing to remember is that if you have a dy problem your limits of integration have to be y values. It also means your integrand needs to be in terms of y. That part is really important and you will have an example of that to watch on your own. So basically then your area of the region is going to be the integral from a to b of the height times the width that's where the f of x minus g of x dx comes from. A really good guiding rule to remember and especially as we continue on into other lessons after this top minus bottom right minus left. You also have a graphing calculator program that's a great tool to use to check your answers. If there are more than one intersection one thing you'll have to notice though and again you'll be watching an example of this. You need to separate it into two integrals. This especially happens maybe when it looks like the curves change position. So you might have to break them up into two different integrals. Just be aware of that. One nice thing though the calculator does. You can type the functions into y equals under any order you want. The calculator will figure out which one is on top. So it knows the order in which to do the subtraction. But most of the time though you'll probably use your function integral or calculate function.