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Point) Let f(r) = Irland g(x) = 22 Tn 2 Find the area of the region enclosed by f (r)ad g(r)Answer:...

Question

Point) Let f(r) = Irland g(x) = 22 Tn 2 Find the area of the region enclosed by f (r)ad g(r)Answer:

point) Let f(r) = Irland g(x) = 22 Tn 2 Find the area of the region enclosed by f (r)ad g(r) Answer:



Answers

Sketch the region bounded by the graphs of the functions and find the area of the region. $$ f(y)=y(2-y), g(y)=-y $$

We're taking interval from 123 of negative to Z X minus one plus three x Yes, like it's us. Negative to the ex away. Q minus X plus three X squared over too. From 3 to 1. If we put in three yet it should be over. Ellen too. Uh, but L b negative AIDS over. I want to minus three plus 27 over too. Plus Q over. Ellen too. Plus one, my ass. Three with you. So this is 12 was one of the 13 10 10 mice six over I to which is equal to I see Go to 1.344

For this problem, we are asked to sketch the region bounded by the graphs of the functions and find the area of the region where our functions are f of Y equals y squared, and G of Y equals Y plus two. Now, for the sketch or for the plot that I have here, I did set X to be the function of Y. So we have y squared or x equals y squared and X equals Y plus two. You don't necessarily have to do that, but we can see that we're going from negative one is when they first intersect or negative one wise when they first intersect and positive to wise when they next intersect. So to find the area will take the integral from negative one up to positive too of y plus two minus y squared dy. So that will come out to y squared over two plus two, Y minus y cubed over three, evaluated from negative one up to two. All that's left is to plug in those values and calculate so one second. The result you arrive at should be nine over to

Alright for this problem, we are asked to find the area of the region bounded by F of X equals X squared and G of x equals X cubed. Uh There's also a plot associated with the problem uh that essentially indicates to us that we are going for X between zero and one. Where over that region F is going to be greater than G or greater than or equal to I should say. So having that to find the area all we need to do is take the integral of F of x minus G of X over our interval. So plugging in what we have, we have that the lower mountain is going to be zero. The upper bound will be one or F of X is X squared and R G of X X cubed. So to actually perform this integration, all we have to do is apply the power rule for integration. The x square will go to x cubed over three. The X cubed will go to X to the power of 4/4. We are evaluating from zero up to one. So that just means that we're going to end up plugging in 1/3 -1/4. Uh To get this, we'd probably or we want to put everything overcome denominator, So that would be 4/12 minus. Um I suppose 1/4 would be 3/12. So we are left with 1/12.

Were given a curve, and we were asked to find the area of one loop enclosed by this curve. The curve is R squared equals sign of tooth data in order to graph this curve to identify what a loop of this curve is going to look like. Okay, let's write this as absolute value of R equals square of sign of two data. So we'll actually drawing two different graphs. One is going to be our equals plus square root of signing tooth data. And the other graph will be R equals negative sign of tooth data. I'll draw the positive. Read the native part in green. Draw the graph line big equals pi over two in a radio access. You see that sign of tooth data? Is that most going to be one? So the square root of sign of tooth data is that most ones we also have That's as long as tooth data lies between hi and zero Sign of tooth data. He's going to be great there in court, a zero and therefore square root of scientific data will be defined. So we're really only considering this top part of the plane for our first function in red. So looking in some points we have that data equals zero. Sign of tooth data is zero. So are you a hero? Data equals hi over four. We have it. Sign of tooth data is sign of pi Over two or one. The square to that is again one. So empire afford trucks and 1/2 between is by 60 angle. Since our Saas deputy that we have this head old like structure Cruz to its maximum values at 30 equals pi over four. And then if it is equal to pi over two, we have squared of sign of pie or squared of zero, which is again zero. So our pedals want to decrease again, back to the origin. And if he's going over the entire range of data since data has between zero empire too. So this is going to be the graph of R equals positive sign of spirit of sign of tooth data. To find the graph of articles negative squared sine two theta well simply reflect these points through the origin. So you still have a point. The origin equals zero and now have a point The third quadrant dictated equals pi over four and we have our third point again that the organ just get another. So we see that this shape has to leaves but are identical and in order, find the area of one of these leaves. Well, suffice to find the area of the leaf for square root of signing tooth data. Okay, So if we say are is equal to swear to sign, to think where Fada has to between zero and pi over two, then we have if we define our to be a function of data which it is and our will always be greater than or equal to zero. And in fact we strictly greater than zero as long as Keita is strictly greater than zero strictly less than pi over two. So you see, there's really only two points this interval. We're going to have to neglect to calculate the area and both these points they're one contributes no area, so we'll have the same area therefore receive it. F is positive function on this interval and because sign of two things is going to be greater than equal to zero in the square root function inside of two data for both continuous functions we have that are is the composition of continuous functions and therefore is a continuous function on this interval. Therefore, in order to find the area of this leaf, you know that this area is well defined. He's given by the Formula Zero empire or two Oh, 1/2 r squared. These data is equal to 1/2 times roll from zero to pirate too. And we have that are squared is equal A sign of tooth data certainly didn't matter which function we chose. Find the pedal. We've had to prove that function was positive and continuous and the other pedal was not depositing motion. So actually didn't, um this is equal 1/2 integral from zero. However two, we actually don't need thing to go. World is taking it a derivative. So this is equal to 1/2 times negative. 1/2 co sign of tooth data evaluated from zero to pi over two. This is equal to negative. 1/4 co sign of pies is negative. One minus co signing zero is one. So this is negative and forth times negative too. Or 1/2. This is our answer. Check our answer. Remember that the area is a positive number since our answer is positive. This makes sense


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