Find the area of the shaded figure, rounded to two decimals: (Assume x = 4 and y = 9.)y yXX...

Find the area of the shaded figure, rounded to two decimals.

Okay, We are going to figure the area of this shape, which is really just to back to back triangles. We actually can see. The two triangles we have are the exact same triangles, the triangles that air 55 in two for their sides. So I figured the yang the size of one of the areas. I could just double it, and I got the whole area. So I'm gonna figure this area and then just double that answer. And I use hair on formula to do this Doesn't matter what side I call a, which when I call B in which one I call see? But I'm gonna call five and five the A's and two will be the sea. The semi perimeter of that triangle is the sum of the three sides divided by two that b 12/2 or six and then using hair on formula for the area, that would be the square root of R s, which is six times s minus a six minus five, which is one times s minus B six minus five, which is one times s minus C six minus two, which is four. That'll give you the square root of 24 which is about 4.9. Remember, we wanted both triangles area together, so we're gonna double that to get the final answer for our area of 9.8.

For the following problem were given to functions. X equals y squared minus four. Why picturing green X equals two y minus y squared. Pictured in blue, we were asked to calculate the area between the two curves shaded in red. Do this. We're going to take the inner girl with respect Toe. Why? So this will be from why equals zero toe y equals three, since that's where the area shaded in red extends from when going from y equals here to y equals three, the blue function appears over top of the green function. Therefore, we will do the integral from 0 to 3 of two Y minus y squared minus why squared minus four. Wife ash on right here Distributing the negative In combining like terms, we get the integral from 0 to 3 of negative two y squared plus six by D y. When taking the integral, you add one to the exponents and then divide by the new exponents so we'd add two plus one equals three and then we divide by three. So it becomes negative. 2/3. Why cubed plus wide to the first power. So one plus one, then we divide by two so six to buy by. She was three, and there's now three times y squared from 0 to 3, plugging three. And for why we get why cubed times negative. 2/3 which simplifies too negative a team. And we get y squared times three, which is plus 27 minus. And then we put zero in for Why? So it's just gonna be minus zero. So becomes 27 minus 18 to given area in between the two curves of nine.

Okay, so here we have this triangle over here, and we're what it looks like two triangles attached to each other because they are because we got 552 on one side and 552 on the other side, so there could agreement by side, side side. So then, if we find the area of one triangle, then we could find the area of the other triangle. We'll just have to multiply it by two at the end. So essentially, we have to find two times the area. Okay, so then the general formula we've got to use is for to find area. We need to figure out what the semi perimeter is, which is one half of a plus B plus c. So a I'm just going to say is five plus five and then I'm going to say see is too. Okay, so then five plus five times two divided by two. What's artistic times? Two plus two. Okay, 55 plus five is 10 plus two is 12 12, divided by two is six. So that's my second character. Okay, so then the area is a semi perimeter six times as minus a, so that's six, minus five times six minus five and then s minus C. So six minus two. So you're taking each semi perimeter is attracting by each side. Okay, so that gives the square root of six times one times, one times for six times, one times one times four is square to 24. Okay, So if I were to break down into break this what's called radical down into simplest forms, we get two times three, which gets six times, two times two, which is for okay. So we could take out a pair of twos for every pair. You pull out one, so you get two square root of two times three, which is six. Okay, remember, since we had two triangles, we got to do two times the area, so we have to do two times two square to six. So that gets the four square to six. So that is my answer. But if we were to do in decimal form four square to six, that would be about 9.7 92 decimal places, so it was 9.77 So that's about 9.80 since it rounds up. Okay. And there you have it. If it's radical or if it's decimal

In the problem we have been given that is Why equal -4 upon texas, choir minus six minus six. So first of all this is written as excess choir minus three X Plus two x -6 Or it is X into X -3 Plus two into X -3 This is X-plus two In blocks -3 No we have This as integration -1, 2 to -4 upon X plus two In two x -3 dx Hence it is -4 upon x plus two And works -3 that equals two. Airborne X-plus two plus B upon x minus three mhm So further we have the value of a equal to four upon five and B equals two minus four upon five. So this is integration -1-2, four upon five And works plus two -4 upon five index minus three index So this is regionals four upon 5, Aaron Model X-plus two -4 upon five. Ellen mode x minus three putting the limits minus one and two. So Father, this is written as Put up on five into Ellen would x plus two Upon X -3 The Elements -1 and two. This gives us 4.5. 16 is equal to 4.5 Ln 2 to the power forward. All this is equal to 16.5 Ln two. So overall we have this as the answer to the problem.