Find Ihe area Ol the region between the Curve and the /-a/isI(/) =16 -/2 frorn 6 Io 6Tne area /5 square units (Type an inleger or & simplified fraction...

Find the area of the region between the curves. $$y=-x^{2}+6 x-5 \text { and } y=2 x-5$$

So this question were given that y equals either the power of negative six sacks and x axis. So the area in the first quadratic between wine the x axis we can right as this interval. Yeah. So now we can make this interval into a limit limit is he goes to infinity. Uh, t zero. You know the power to your six Anstey, which we can then rewrite. This portion is negative. One over 60 the power of negative acts. This is Europe. So up here and now we have the limit I see goes to infinity one sex minus 16 year. The power negative Cheap. Which one we evaluate is gonna give us 16 which means the area and this washing is one sex.

In probably 37. We want to get the area enclosed by these two parabolas. Its first sketch, These two problems. To visualize the shaded area, we substituted by different values off X to get boy for the first problem when X equals zero, we have y equals five x equals one we have war equals tu minus six minus four plus five equals one When why equals two. We have to multiply it by four She's eight minus 12 minus four plus five equals one. We have zero and five zero and five one on one two on one. This is a parabola for the first girl. Something like that. We don't know it goes below the X axis or not. Never mind. Let's go. The second parabola when X equals zero we have one equals minus 15. When X equals one we have X, we have y equals one plus 67 minus eight. When X equals two. We have boy equals for lost 12 equals 16 minus 15 Swarm When X equals three we have oy equals lying plus 18 violence 15 You have all equals 12. Let's start with the first point for the 2nd 0.1 on minus eight on I minus. It's something like here. 211 here. Three and the 12. Something like here. Then we have the problem. This problem because something like that And if we continue this problem, they will intersect at some point. That's a good the ex corded. Off this point, we can get this coordinated by equating the Y Coordinate off both graphs. We have to X squared minus six X plus five equals X squared plus six x minus 15. By solving this equation, we get two points. This is the first point we know it. It's X accorded equals two, and this is the second point. Let's get it's extraordinary. We subtract X squared from both sides. X squared minus x x plus five. Equal six X minus 15 We subtract six x from both sides. We have X squared minus 12 x last five equals minus 15. We add 15 to both sides, then X squared minus well X plus 20 equals zero. You want to get two numbers. Their multiplication is most of 20 and there some is minus 12. The two numbers is minus three and minus four. And there there's some is his minus 12. We have 20 can be to beytin. The two numbers is two and 10 x minus. Stoop The two numbers the minus two and minus 10 x minus two. Want to blow by X minus 10 equals zero. The first point is at X equals two, and the second point is at X equal. Stine. They meet at X Equal Stone this point as accorded equal stone. Never mind about the sketch, and it's a scale. It's not to skill. Let's use the definition off the definite integral To get this area. The area equals the top function a definite integral off the to function, minus the bottom function. That function is the second parabola here, and the Balkan function is the first bar apology. Then we have the definite integral. Off the second problem. X squared the blast X X minus 15 minus the first problem two X squared minus six X plus five The X We integrate for the bones off the area. It starts here at X equals two and ends at X Equal Stone or the solution off this equation. We integrate from 10 from 2 to 10 from 2 to 10 that's integrate X squared minus two X squared is minus X squared plus six x plus six x plus 12 x minus 15 minus five is minus 20. The X from 2 to 10 equals minus execute, divided by three plus 12 x squared, Divided by two When I was 20 x, we integrate from 2 to 10. We started by substituting by the upper bound with service Uber X equals 10 we have minus 10 pube by three plus six multiplied by then squared minus 20. Multiply by 10 minus. We substitute by the lower bound minus toe cube by the by three plus six. Multiply by two square minus 20. Multiply by two Evaluating the first term. Gives minus 1000 by by three plus 600 minus 200 equals 205 by three We have minus. Second term is minus eight, divided by three plus six months by before minus 40 equals minus, and we have a minus sign. 56 divided by three equals 256 Divided by three This is the area bounded by the two given problem in the problem

So we need to find the area enclosed by the colour White calls to a group of more X and fire wire was to express six. So to find this area, we know that's why it goes to the root of more X here. Models represented X cannot be negative and we know inside the room if we take Xnegative becomes complex number. So we'll take X positive value and why it was to protect us. If I plant the diagram, we have a call like this. So this is why it was, too. The route eggs or root marks. It doesn't matter now. We have our functions since we are only positive. Corporate now five y equals two x plus six. So we need to find the intersecting point of these two curves. And to find intersecting point, we have white close to Route X 90 square. So you get wise where it goes to monarchs, right? And this says her ex will be positive. And of course it will be since Wise Square so we can directly that was very close to X. And then we have another equation which is five y equals two x plus six so we can substitute wide square here. Or you can square this. Who will get 25 wide square is X plus six old square and then you can substitute your eggs. So this becomes 25 eggs has X plus six whole square. And to find the intercepting point, we have to open this bracket and solve it. So we get 25 X equals two X squared plus 6 to 12 X plus 36 therefore we get X squared minus 25 minus 12 or 12. Minus 25 is 13 X plus 36. So we get our intersecting point on solving at X equals two x four and nine. So for exit equals to four people substitute here we get Why so 44 we get y s do. And for 99% 15 15 by 53 So there are two intersecting point. Great. That is that Two comma four and three comma night. So when X is four, Let's see, here we have four. There is intersecting point in y axis which has value of two. Okay, And for X equals to nine. Because when you are mhm. Okay, so now we have obtained the region that we need to find which is this part? Yes, it is visible to you. You can shade it so this area we need and this area is bounded between two curves. Sorry, this is a point. But the line passes through these two points. So the line of fire way calls to six will be like this. And the region that we need to find this only this part. Okay, now let's calculate the region. So we have our starting and ending point from so area will be now here limited the left and right boundaries. So let it start from four and at nine. And the curves are upper boundary and lower boundary. So upper it is bounded by white goes to root eggs. So why do that? Is rude checks and lower is vulnerable. Fire y equals X plus six, which is X plus six bite. Right. So this is our Yeah. Would you be now? It will be positive so we can take more here. Okay, so let's integrate. Celotex Integration is expressed to three by 2/3 by two Minus year one by five Explosive integration is X plus six square or two. So let's substitute our limit for 29 So when will substitute? Nine? This becomes nine. Rested three by two for a group of 93 and the Cube is 27 over. Three times two minus one by 10 When we substitute 99 plus six is 15 15 square is 2 20. Fight. So this is our upper limit minus lower limit, which will need to buy three. When we substitute four, we get four rotors to Cuba's eight minus one by 10. Four plus six defendants where? So I'm solving this equation or expression. We get areas one by six Unit script. Thank you.