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Problem 59 Hard Difficulty

Find the values of $ c $ such that the area of the region bounded by the parabolas $ y = x^2 - c^2 $ and $ y = c^2 - x^2 $ is 576.

Answer

$c=\pm 6$

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Video Transcript

So we want to find the value of see such that the area of the region bounded by the parabolas. Um, these two parabolas that's listed here is 576. So the first thing we want to do is find the point of intersection between these two guys so that we can get a feel for what region we're actually looking at. So in order to find the point of intersection, which I've labeled P o I let's set, um, why equal why or x squared minus C squared equal to C squared minus x squared. And then we do the the quick algebra at X squared to both sides ad C square to both sides so that we can just get the exes on one side and the constance on the other side. So those cancel and we've got that X squared equals, in fact, two x squared, because to C squared the twos, divide each other out, and then we're solving for X. So X is equal to plus or minus the square root of C squared, which is equal to plus or minus C and then playing see back into either one of these functions, we can find the value for why and we know that plugging in plus or minus C will give us c squared minus c squared, which is why equals zero. So the two points of intersections that we have since we have to see values are minus C for X and zero for R Y and positive psi for X and zero for our way. So those are our points of intersections. So let's graph thes proble ce over here so that we can get a feel for what we're actually looking at. So roughly the first guy is a parable a that is pointing upward and shifted downward by C square units. So it is something like this on dhe. I'm not gonna be super precise. I just want a general feel for what I'm working with. And then the next problem which I'm gonna label is green. That is a downward pointing parabola shifted upwards C square units. So something like this and then we know that our points of intersection are at, um when Why is zero? We have negative C and positive. See, So this year is X equals negative. See? And this is X is positive. See? Okay, so we want to find the region. Um, that is enclosed between the blue and green curve. We want to make that area 576. And we want, um, to find the sea values that ensure that this whole region is 576. So what we need to do stake an integral. So we're going to set the area, right? So an integral is area. So we're gonna say the area which we want to be 576 is equal to the integral from, um, our left, um point of intersection to our right point of intersection. So X is negative, see to see, And then we subtract the bottom from the top. So green minus blue, which will be C squared, minus x squared minus. So that C square minus export is our blue, and then our green is C squared. Um oh. Sorry. Said that backwards, um, c squared minus expertise are green. And then our blue, which is on the bottom, is X squared minus c so minus X squared, minus C squared DX. That gives us the interval from negative. See to see of distributing that negative through we have two C squared minus two x squared the ex, which is equal to now. Remember that sea is a constant. So when we're taking the integral of this, this will be to see Squared X because this whole thing is just a constant. It stays there minus 2/3 execute evaluated from negative CDC and then plugging that in. We have to C squared times, see minus 2/3 say cute minus plugging in the lower bound gives us a negative to C squared times See, pulling the negative out in front and then plus 2/3 see cute because of the negative they canceled to a positive and then simplifying This gives us I'm gonna make a common denominator. So, um, common denominators. Three So six over three c cute, minus two over three C cubed, plus six over three c cute, minus two over three c cute. And when we put all that together, we get eight over three CQ. So since we're trying to solve for C. U, we have 576 is equal to eight over three c cute and we just do the algebra we'll supply above and below both sides and eight goes into 576 72 times and we're left with 216 equals C cute and then take the cube root of both sides and 216 is the same as six. Cute see, And so that's equal to six, of course, so C equals six. But then we have to go back and were asked the question to find the values of see such that the area of the region bounded by the parable is is 576. And looking at our equations of our parabolas here we have C Square's going on, so C can be positive and that will still satisfy. The problem is, um, it will still give us the region between those parabolas. If we insert a positive see will still give us 576. Likewise, if we insert a negative, see into these, um, equations for the problem was, the area that's enclosed between the two will still be 576 because that square it cancels out the negative, so see, in fact, can be positive. Six or negative six so see is equal to plus or minus six. It will still satisfy, um, having the region between those two paralysis equal to 576