In this video, I'd like to give you an example of a region that is bounded by several curves. There's many different ways to define a region, and of course if you can do it with two curves, you can certainly do it with three. Let's jump right to it with an example. This is going to be pretty typical of what you're going to see as well. If I have a region, so it's here, I'll set it up. Find the area of the region bounded by the following. Let's say y equals 3x plus 4, y equals x squared, and then y equals minus x plus 2. If you think about it, and this one is, three sides, it's not going to be a triangle because of the parabola here defining this thing, but it's a three-sided region. You can have four-sided or five-sided, but on any thinking as you know just two and three is probably as bad as you're going to see it. If we can do this one, you could probably do anything else you get. The key to this one, as with the rest of them, these are easy enough graphs, now before I go ahead and try to start figuring out some bounds and all this other stuff because you notice the bounds are not given, let's draw a picture. Let us draw these things carefully, and as best we can draw them to scale. Y equals x squared, okay, so that's the good old parabola. It goes right to the origin, make it super pretty, try not to draw the world's smallest pictures, nice and big, so we can find points of intersection. The next thing that I would draw, I guess is the line, let's do 3x plus 4. That's positive, so going upwards, if I plug-in 0, I'm going to get 4, so it's going to cross 4, I don't know where to put that thing. Let's just sketch, something like that. It crosses at 4 and off it goes, and they'll touch up here and that's fine, but there's another curve coming. Let's just label this thing, so y equals 3x plus 4, and I guess I could plug in where it crosses. Well, I don't know. That should be good enough. If we need to add more to a picture later, we'll do so, but right now we're just trying to get the gist of it. What's the next one? Minus x plus 2, so this is the line minus x, the diagonal one with negative slope, and then I shifted up 2 and that line is going to do something, it'll cross this way. This is y equals negative x plus 2, so a line negative slope shifted up 2 from the origin. I have my region, and as promised it was, I don't know what this thing is, like a parabolic slice of pizza maybe I don't know. It's not as symmetric as my picture might perhaps draw. If you want to see a better picture run over to Desmos or something like that to graph this thing. But you get the idea at least of what side is what and then you can see what's there. Once again, as promise, the bounds are not given. Again get used to it, and then this way it's only a pleasant surprise when they aren't given. The bounds are not given. What do we got to do? We got to do algebra. Well, how do I find where the bounds are given? Remember that the bounds are given where the curves intersect, so what's a and what's b? I got these two curves here that talk. On the first one, the line 3x, so let's find the lower bound here, find a. The line 3x plus 4 intersects the parabola x squared. We have to do some math, we can do this. We get x squared minus 3x, minus 4. When I do that, that's a quadratic, it's not terrible, why? Because I can factor that, I don't have to do quadratic formula. This becomes x minus 4 parentheses x plus 1 is 0, and I get two solutions. I get x equals 4 or x equals minus 1. Now, this is where the picture is extremely helpful, which one do I want? I only want one of them. I'm trying to find one number, but I have two choices. Well, I know that the left bound is to the left of the x-axis, so of course the one I want is the negative one, so x equals negative 1. The other bound is going to come from the intersection, so let's find b or intersection. This comes from the parabola once again, x squared and it's going to intersect the line with negative slope, negative x plus 2. The more time you put on the diagram, the easier this is to follow. Same thing let's move everything to one side, x squared plus x minus 2 is 0, that will also factor. If you stay that long enough, you'll see it's x plus 2, x minus 1 equals 0. Therefore you get two choices again, quadratic two solutions, you get x equals minus 2 or x equals 1. The picture helps us determine which value we want, we want the positive value, so x will equal 1. Now we have our bounds, so we did algebra, yeah algebra. Now we look this thing, does it across anywhere, what's going on? This is interesting because I have two curves, it's getting complicated, we have three curves but two curves play a role here, and there's two regions. There's a region here to the left and then there's a region to the right. Where is this thing? I have a region to left, that's our first region, and then I had the region to the right, so we can call this A_1 and A_2. This cannot be captured in its form as a single integral, can't do it. You need two integrals, so we have our A_1 as our left side, and then our A_2 as our right-side, the curves change. Now we have another problem because now I have to find another intersection point. I'm going to have two integrals, I need another intersection point which we can call c. This is where the two lines intersect, that's not that bad, 3x plus 4 will equal minus x plus 2, 4x equals minus 2, so x equals minus a half. I have to set this up, finally, all this algebra, let's set up the integral. The area is going to be the area of the left side, plus the area of the right side. To do that, be careful, I'm going to go from a to c or on our case minus 1 to negative a half, and it's the top curve minus the bottom curve. The top curve the line with positive slope 3x plus 4, minus the parabola dx. Then I'm going to add that to the other region, which is from negative a half all the way to 1, yes, good. This is the top curve minus the bottom curve, now be careful it's a different line, so it's minus x plus 2 and then I also subtract the parabola, so there's a lot going on here. There's three curves, two regions, no bounds. These integrals though at this point you can almost breathe the sigh of relief if you're pretty much done, and now you just have to be very careful plugging in, it gets a little arithmetic heavy but you can do it, so I leave that to you as an exercise. At this point, you have to work out two integrals, so go off on the side, do the first one. If you do the first one by the way, and check you should get 7 over 12. If you do the second one correctly, you should get 27 over 12. When you add these things up together, so you get 34 over 12 and reduce, you'll get 17 over 6. Work that out and check your answer here. But be aware of these regions bounded by several curves, there's nothing stopping me or the book or somebody adding more, but they tend to get a little tedious, so two or three is usually the worst, you'll see. Good job on this one. See you next time.