Hello. In the previous session Because there are two types of integration problems I mentioned. Bi species of the region with a variety of functions, You can define the number and integral with Do you want to write. This integration will be most noticeable when writing which subjects to be more suitable and limits of can be written correctly. In the second type of problems directly is not allowed to function. But, and that was an integral The boundaries and sequences integrally entergal is given. We want to find in here. Typically, as in this example two with the total entergal Once you understand the integral of a given region After one integrals possible to express the same result, can be. Now we will see an example here. As you can see in here directly undefined, but indirectly with the limits of integration and again as a useful information also given the order of integration. Now how do we solve it? Not too difficult. Wherein to determine the area before let's set limits. There are two types of boundaries as you can see. Type of x is equal to a limit x is equal to zero y is zero, y is a zero gene is repeated. x is equal to a constant as x is equal to two with the values given. They are given here in the second row. Another type of x y is borderline. y is equal to x in this example two minus variable given limits. Now this information has been gathered bi After this general obtained with a variety of border crossing points It is. Here, for example, y means x is equal to zero mean axis. means that y is equal to zero means that the x-axis. x is equal to the vertical a x is equal to one means a line. However, the more an account of the most vulnerable that requires this with different boundaries, with variable limits given. Curves, but the curves also confirms here would have to find the point of intersection. See here is the line y equals x. BI also have a line of y equals x minus two. They should intersect. They intersect a region already closed can not occur. We are looking at their point of intersection. That means finding the intersection of y the To synchronize the eşitleyin y de x is equal to two If this party had minus two of x x x We find the x equals one equals two. whether this first one when x is equal to Let's find the function you want from the second Let's find of the function y equals a 're getting value. An intersection point of a means of point. Let's call him. Now to understand the region will continue. Here again, month, she wrote, and the resulting integral we We wrote the first but very easy to obtain information There are other points that are bi. X is equal to one of them becomes zero bi If you look here y is zero when x is equal to zero 're getting that. Also the starting point easily we see. here, of course, when x is equal to two There are several spots x is equal to If you find the time to be here two years equals two, but you can find them You can try them all. x equals zero, y is at bi than two There are solutions. This gives point b. Now we draw what we find once bi y the first line is equal. This is the first bisecting angle. In the second function, a second border the x and y is equal to the slope of two minus x minus this a which x is equal to two passes through zero right this second. See the intersection point of these two a. We found it a little while ago already. Meanwhile the b point and that point We know. b point here. Two and a zero point. Already y is equal to zero. See where continuous a bottom border. Y is zero means that this x-axis is going on. But even so, this intersection göremesekte two of the points from point zero zero zero and the We're creating a triangle. Of course I need to check in bi really whether this is true. Indeed if we look at the integral over there. Kept constant for x to x years ago from scratch going. So we provide it. Then x goes from zero to one. So, our main area of the first sub- This region gives integrally on first term. The second term is equal to zero if y y equals two minus x to arrive. This negative sloping downwards on the right. Yes, we see it here. Similarly the x is equal to one to two up scan the second sub-region have gone, we're pretty much covered. So here we are providing this idea. Now here we can see easily that selection of two choose a suitable one that leads to the integral do not choose. Instead, we fix x on y If we go see a kind of all regions with boundary conditions able to scan. Now, where x is equal to y y is equal to x we can get. See here for the first year is equal to limit x line're starting out. Here we fix the x to y is equal to the second year we're going to the limit. x minus y equals two. As you can see in this way the whole region able to scan. And we look at the limits of y a point in time this year the biggest scratch value of y in this region. This area has been shut scan would have. As you can see we understand the region. Once you understand the same region the integral will result in a single integral We have determined. As a second example:This is a bit more may seem complicated. Yet here are the same types of structures. A y and x on the fixed values is there. Here only on the constants y see that. y is equal to zero. Y represents a there. y is a repeated and y is nine going on. To say that we scratch on y We're going on a region far. On a region to more than nine that goes from zero to nine years a place. When we look at the function variables x is equal to the square root y y and x is equal to the square root of minus there. We obtain the equivalent thereof frames Taking y equals x squared. Bi parabola this means that geliyo limits. When we see here again a parabola here we see the branch. In addition to this, such a straight Gene There equation. That's right, we arrange according to the equation y here consists of two x plus three. Y is again zero, one and nine values here are writing. Now, at the intersection of the boundary lines that requires some interesting account. The intersection of this parabola that's right. They say the intersection of y means to synchronize. As you can see x squared minus two x minus three equivalent is equal to zero. This is a second-order power function. We know it to take root. Here is two half formula We can use. This is one in a plus or minus the square root of the minus the square of this Multiplying the number one. Here you will of course minus the left-hand side after passing. So the bottom plus the square root of enters. So. Roots of a plus or minus the square root of two to occur under four. Already a number of figures such beautiful you might be problems we've designed it. Now if we look at the value here before the Let's take small. A minus is a minus two from x. One plus two is three from x. If we look over here a while x minus y would be the one. Similarly, on the other intersecting boundaries has calculated If we had a yield of x minus y still a that's as it should be. This one is going to provide. X is equal to the time we came here more than two years has nine. Yet as x three that we provide When going here for six. Giving six plus nine three genes. Now I still have zero zero. Y when y is zero, because here is equal to zero, we find. wherein Y when y is zero equals We find but maybe that three of our region not in. Maybe not a limit. Therefore, we can save him but not much there may be no benefit. here again, when Y represents a We find a value. From their meaning in the next number to be understood. See it here again, these two integral Let's write. These are the results we found. When we look at the easiest y is equal to x the parabola y equals x squared square. two x plus y equals three it's true. These two cut-off point here As you can see in that a segment of a parabola y is had. Here, a value of minus one and plus We find for x. They also seem already here. Minus one. After the cut-off point x in y equals nine 3üç. In all these things we do not provide. Now, months, and again, let's look at supply. The first integral of the parabola from these branches, the wherein y is equal to x is equal to the square root of x minus y, but we see that mark. This negative value plus the value of x is going. y's kept constant in the horizontal mean We are moving this year to get that fixed In the integral. We're doing the integral over x. Reset really took one year of their territorial boundaries. The second sub-region now than when y In nine going to the outer boundary. We provide it. See also in internal borders in the horizontal strip y constant we hold the right of x over value When we calculate the x is equal to y minus three half. Here again, the square root of x to the value of X, y. X plus or minus, of course, but that could have been negative values are outside of our region. In this way we provide it for him we see. We now recognize that vertical with strips so we start with x to y held constant What if we see the first integral x to y held constant integral When we get out of the parabola y equals x squared equals two x plus y from 3're going to. wherein the values of x less than three, I'm sorry you're going less than three b. These are the largest value of x. As you can see we know of. It recognizes the hard work in side after this to enter single-storey, single integral to translate It is possible. The result of these two integral same value to be given to work with this second We would prefer. Because one will calculate the integral. Not given here because it is important for here give a f integral to it again as measured by a What is important is to understand the region find out. Learn how to write and borders. A third example. I'll leave it as homework. Y refer here again limits is there. x square root is going on. First year on the integration. So the fixed x, y, on the one are integral. Here, the integral over the year again. As you can see here certain limits. Now we can easily see where the x y equals the square. See also includes these two boundary y equals x The square parabola axis X axis a parabola. As you can see, we can draw it. The previous example is a function It looks like a little. There is a slight difference. This is y equals x minus two. A slope. x is equal to y minus two is equal to zero passing or y is equivalent zero x equals two going. As you can see this is true. So it's true that our region of the parabola consists of the intersection. If we fix the y on x before the See if we go with integral to the point of intersection of a parabola We're going from branch to branch. These limits. The square root of x. But after this intersection directly from the parabola We're going. For him to write a second integral is required. However, in the region y fix the first integral x, x on If we take here As seen in this single expression integrals able. If we let in because where these area to the right of the parabola parabola We're going to the right. However, from a branch of a parabola here If this is the first branch here in the sub-region we went directly to a branch of the parabola for consists of two different host our region. Recommend doing this as an example I would. Looking at an assignment. Once we have understood them an easy things to come. On the x and y refer here again There are constant values. There is a limit of the variable. In general, we see immediately that y have equal x. This is a right. First bisecting angle. If we take the square of this equation x squared plus y squared equals four. So the radius of the coordinates of the center of two The circle in the center of the team. The intersection of these circles say that this is true is there. But that does not cover both the immediate area. But here we see that y is equal to zero is there. y is equal to zero when you bring in the closed is going on. What do others works? Nonsense, but not others defines in primary No tasks. x is equal to zero. E already it's true that the x-axis Since the intersection in our region. Then x equals two, this circle's x axis the point of cut. The critical point of the square root of x the point where the two. Indeed, these circles of this hacking If we look at the root of two is cut we see. This gives the cut-off point. Thus, we can provide this integration. After providing them work to be done integral order we fix before changing y x integral on the letter. As you can see on the integral of x x Y value This is fixed for x is equal to y the right of the equation. On this circle x minus y equals four go to the square. See all regions directly in the circle head on With only one team going to the limit circle type We can define the region. In the second integral of y in the integral outer y is the smallest non-zero value maximum value at the point x in this section square root of two, y is the square root of two turns. You can easily provide. You can control. I'm waiting for you to find such a response. Now we'll take a break here today. However, to say the next section I want. Why is the moment to make accounts I'm waiting for? Once things that work. Integrals give you a random team of them Calculate the mean, rather than your business from a number of will help both prefer to have realistic examples I've also mathematics Thinking in terms of infinitesimal with an important experience. That's the moment when the accounts statements of torque that's likely to find significant experience with infinitesimal are able to earn. Why moments? Because the important moments in the physical sciences the example also used in the science of statistics sizes. These are reviewed in the next session We will spend some calculate the moment of making additional with the concept of the infinitesimal Thinking our experience will improve and the double our ability to calculate the integral story will solidify. Bye for now.