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Solve the given problems by integration.Find the area bounded by $y=\frac{2}{\sqrt{4-x^{2}}}$ and $y=\frac{4}{4+x^{2}}+1$ Use a calculator to find the points of intersection.

Calculus 1 / AB

Chapter 28

Methods of Integration

Section 6

Inverse Trigonometric Forms

Integrals

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So for this problem, we are going to be finding the area between our two curs given to us by these expressions. And so our first up in any problem where we're finding the area between two curves is always going to be graphing our functions A regardless of whether or not the problem tells us to. Um and this is just good practice to make sure that it's much easier for us to essentially just determine which function will be on the top and which will be on the bottom, which makes it much easier to set up our integral. And so when we craft or two functions, we can see that we have this red one which is the right half of a circle. And our blue one is a hyper bella. So, essentially, if we find the area in this section, since both of these shapes were going to be symmetric, we're just going to most by that section, fight to in order to get the total area that includes this lower section as well. So our first half of this problem is going to be finding our point of intersection at the top here. So we can see that our circle is going to stop at going toe left at X is equal to zero just because the fact that this is only the right hand side of our circle and so essentially just from here. But we're going to use our graphing calculator to figure out this point of intersection and we do that we find that our X is going to be equal to 1.2 to 5 other. So now if you go and start setting up our integral, which we know is going to go from zero through 1.2 to 5 and our top function is this red one. Um however, our function has been to us as X in terms of white and because we found or ex coordinates here for our boundaries, we're going to want to rearrange us to be Wieters of X. So if we square both sides, this function we get X squared is equal to four month's Y squared Now and then with a little rearranging, you will get why squared is equal to four month's export. So we can write this as wise equal to the square root of four minutes. Export which will go inside of our internal. And then we're just going to subtract our second function, which is why I squared is equal to one plus X squared so we can rewrite this as wise equals the square root of one plus X squared. And instead of having both the positive and negative here, we only want the positive version because we're just finding the top half here. Comdex. And so now that we have are integral, completely set up, all we have to do is evaluate this, using the techniques of integration that we learned in Chapter one. And so once we do that and then apply the fundamental theory of calculus, we find that our area for this top section is equal to 0.802 And so to get our total area, we're just going to buzz by two. And when we do that, we find that the area between our two curves is equal to one plate 605

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