 This lesson is on area between two curves. In previous lessons, you found the area underneath the curve and above the x-axis. In your lesson, you should have gone over the different types of areas other than the one that just lies above the x-axis. This video will concentrate on those other types of areas. But first, let us review area above a curve where the curve is positive. Determine the area under the curve of y equals sine x from 0.25 to 0.5 radians. And we know that the curve of sine looks like this. What we are looking for is a portion in here that looks like that, and we're looking for that area under the curve. So a very simple formula, area equals. And if we did summation of Riemann sums where we took the area and broke it up as we did before with different areas like this, we would get the summation formula with the limit as n approaches infinity of the summation from i equals 0 to n of sine x sub i delta x sub i. And that would give us the integral from a to b where we begin, where we end on our graph of sine x dx. And in this case, our a is 0.25 and our b is 0.5. So as we think through these areas, we have to think through Riemann sums and how they actually work in building the area under the curve. So once we get our integral from 0.25 to 0.5 of sine x dx, we just integrate and put the numbers in our calculator because these are not exact values for sine or cosine, and we get the answer of 0.091. You can do this by hand and say the integral of sine x is negative cosine x, and we're going to evaluate it from 0.25 to 0.5. But either way, put it in your calculator or do it by hand. It doesn't matter. You should get the same answer out. But let's go on to extending our knowledge. What other types? Well, we're going to learn how to find the area between two curves. And we're going to learn how to find the area when a curve crosses the x-axis because you'll have negative area as well as positive area in this one. We will also learn to find the area between two curves that cross each other because one is lower at one time and then higher at another time. And we will find the area between two curves that are turned sideways. And keeping them in x's may not work as efficiently as changing to the axis being the y-axis and looking at them in y's. Find the area when a curve crosses the x-axis. Find the area of the region bounded by y equals sine x from 0 to 5.5. Well, this time we have our sine wave. It goes up. It comes down. It goes up a little bit more, about 5.5 radians. So we want the area above the curve here and below the curve here. And you have learned the area below the curve is negative. So what we must do is find out what this point is so that we can take the absolute value of the area below the curve and add it to the area above the curve. And we all know that this point is when x is equal to pi. So when we break this up, we will have the area equals. We're going to go from 0 to pi of sine x dx. And we're going to add to that the absolute value of the area between pi and 5.5 where we end of sine x dx. And calculating that out, we will get 2 plus the absolute value of negative 1.709. And that becomes 3.709. Again, when we cross the axis, we have to account for the fact that this is negative. So we separate the two. Find the area between two curves. Find the area of the region bounded by y is equal to sine x, y equals cosine x and the y axis. So if we draw the curves out, and it's always smart to do this. Sine x looks like this. Cosine x comes down like this. And we can see y axis is here. So this has to be our region in here. So where do they cross? Well, let's see. Sine x equals cosine x when x is equal to pi over 4, something you already know. So the only thing we need to do now is set up our area formula. So area equals the integral. Now when we look at these, we see that the area under the upper curve is this area in here. All the way down to the x axis. So under the upper curve would be cosine x. So we'll say cosine x. We are going to subtract from that the area under the smaller curve, which is the sine curve. And that's this area in here. And what we will be left with is the area that is all cross-hatched. So it's cosine x minus sine x. Of course, that's dx. And it goes from 0 to pi over 4. Taking the anti-derivative and evaluating it, we get an answer of 0.414 for that particular area. Again, upper curve minus lower curve. And when we go on to volumes, you will find that you will be using this idea much more in doing those volumes in volumes of revolution. Let's go on to another type. Find the area between curves which are not positive. They go from positive to negative. Determine the area between the two curves. Y is equal to x squared plus 2. And y is equal to x minus 1 over the interval negative 2 to 2. Well, again, these are fairly simple to draw. x squared plus 2 looks something like this. And y equals x minus 1 looks something like this. And we want to go from a value of negative 2 to positive 2. Now, if we think about this, if we take the area under the curve of y equals x squared plus 2, we'll have this area in here. If we take the area under x minus 1, it will be negative and it will be here. So what we do is subtract the two areas. So it looks like this. The interval is negative 2 to 2. Upper curve x squared plus 2 minus the lower curve, which will make that one positive. And that one is x minus 1. And all of that is a dx. And if we want to go on and actually work this one out, it would be x cubed over 3 minus x squared over 2. And then doing the 2 and the 1 would be plus 3x. And we will integrate that from negative 2 to 2 and find out the answer is 17 and 1-third. You can do it by hand. You can do them on your calculator. As long as you stay efficient in doing your derivatives and integrals, you will be fine in doing it either way. Let's go on. In this one, we're going to find the area of the region bounded by y equal sine x and y equal cosine x from 0 to 3. And 3, as you know, is less than pi. So if we sketch these out, we see that the sine curve doesn't quite reach the x-axis. The cosine curve comes down and goes there. So we are looking for the two regions. The first region sits there. The second region sits here. And if this were a 3, the upper curve would be cosine at the beginning and then sine once it crosses over. So our area would be the integral from 0 to where they intersect, which is pi over 4 of the upper one, which is cosine x, minus the lower one, which is sine x. And we're going to dx that. And we're going to add to that from pi over 4 to where the two curves end, which is at 3 and switch the two around. We'll have sine x minus cosine x dx. Integrating those, adding them together, we get the royal sum of 2.677. You may, again, do these on your calculator or not. Do them by hand, whichever one you want to practice. Let's go on. Now we go to a different idea. Find the area between curves that are turned sideways. Find the area bounded by the region. x is equal to 2 times y squared minus y cubed and the y-axis. If we sketch this one out, it looks like this. So this time our merimon sum is taken from the y-axis and everything is done in y's. Not a big deal. We just have to find the beginning point on our y and the ending point on our y. And we find that x is equal to 0 when y squared times 1 minus y equals 0. We don't have to worry about the 2. And that's when y is equal to 0 or 1. So our area will be an integral from 0 to 1. The only thing we have to do is put our function in 2 times y squared minus y cubed dy. It seems strange to be in y's if you want to switch the variable. Fine, but you can just keep it in y's. And ultimately you will get an answer of 1 sixth when you go through all the mathematics on this. That one seemed easy and obvious to do it in y's. Let's look at another one. In the area between the curves, one of which may be both the larger and the smaller over an interval. So this is our problem. Determine the area enclosed by the curves y is equal to x minus 3 and x is equal to 4 minus y squared. If we sketched out the 2 curves, we will find that they look like this. The x minus 3 looks like this. And the 4 minus y squared actually comes around like this and goes to a point on the y axis. So if we try to do this in x's, you see at this point we have the x's equal to 4 minus y squared both as a top curve and the bottom curve. We can do it that way. It's a little bit more involved. If we look at it from the y perspective, we see that our top curve is always the 4 minus y squared and our bottom curve is always the x minus 3. So it would be a lot easier to set it up this way. Well first of all we need to find out intersecting points. If we make these two equal to each other, so we have to change the line around. So x is equal to y plus 3. And if we set that equal to 4 minus y squared, we ultimately get y to be 0.618 and negative 1.618. So we do have those two values for y so that we can set up our integral. We see that the area is equal to the integral of negative 1.618 to 0.618. The upper curve is the 4 minus y squared. We're going to subtract that lower curve which is y plus 3 dy the whole thing. Put in our calculator since we have these values for the integral and we will get the royal sum of 1.863. These are all the different kinds of areas between curves that you could possibly have. So work on these, study these examples because they do address every single type. This concludes our lesson on area between curves.