 Hi everyone, I have here for you another example of finding the area between two curves. So here we're going to find the area of the region bounded by the graphs of f of x equals 2 minus x squared and g of x equals x. So let's take a look at them on our graphing calculator and just go ahead and do zoom 6 to graph them and this is really your classic area between two curves problem. Now you'll notice one thing that was not mentioned in the problem are any kind of boundaries or limits of integration. Those were going to have to get from the graph so you'll need to go ahead and find those points of intersection. I shall leave that for you. I've already done it and I know what it is so they should be negative 2 on the left down in the third quadrant and then x equals positive 1 up there in the first quadrant. Now think about the orientation of your representative rectangle. If you want that rectangle to hit both curves it needs to go vertically. Therefore it's going to be a dx problem. Therefore your limits of integration need to be the x values from those points of intersection. But please do try it to make sure you get x equals negative 2 and x equals positive 1. So let's go ahead and set up our integral. So our area then will be the integral from negative 2 to 1 of remember the 2 minus x square curve was on the top. Remember top minus bottom minus the bottom curve which was x. So we can go ahead and do our anti-differentiation and evaluating that from negative 2 to 1. So you can do that by hand. I'm going to go ahead and use function integral on my calculator. So math 9 from negative 2 to 1. So I'm going to do this a little bit faster. I know that my upper curve the 2 minus x squared is what I had under y equals under y1. So if you hit your vars button go across the top to y vars and under function you can pull up y1 that way. So just a little bit faster saves you a little bit of typing. Then we can do the same thing to pull up y2 which is where the other function was. Just hit 2 so it looks like that. So the answer we get a nice 4 and a half. So once again we could check that by using that area between program. So let's get that a try. So our lower boundary in this case is negative 2. Upper boundary is 1 and voila there's the answer. Again a really great way to check your answers.