estimate-the-value-of-k0k1-so-that-the-region-enclosed-by-y1-sqrt1-x2-yx-x0-and-xk-has-an-area-of-1-

Question

Estimate the value of $k(0<k<1)$ so that the region enclosed by $y=1 / \sqrt{1-x^{2}}, y=x, x=0,$ and $x=k$ has an
area of 1 square unit.

Gregory Higby · Accepted Answer

eso this is Ah, an interesting problem. Um okay, so this is the functioning. Gave you the 1/1 minus x squared. Um, this is the other equation y equals X Sharon y equals. That&#x27;s the deal. I equals X. We know that we&#x27;re going from zero, and we&#x27;re going to X equals K and the premises. We want the area between the two curves to equal one. So it&#x27;s pretty straightforward once you see the graph that you sang it up from zero to K of the upper function minus your lower function, Um d X equal one. So the only issue that you&#x27;re gonna fall into is what C and I dreamed of this. And a quick Google search, if you haven&#x27;t learned it yet, is that is the inverse sign of X the and a review of Xia. I&#x27;d want to your exponents divide by your new exponents from zero to K still equals one. So now you just have to plug in. In were sign of K minus one half times K squared. Andi, I did this in a previous video that I&#x27;m not gonna worry about. Science, Enver. Sign of zero assigns the Y coordinate. So right here. What&#x27;s the radium measure? It&#x27;s still zero, and I&#x27;m plugging in. Zero here will zero square to steal zero. Subtracting zero doesn&#x27;t change anything. Um, so the main thing now is to go to a graphing calculator to figure out where this K value ISS Um, what I would do is ah so, um goto graphing calculator and maybe just type in cause you consign both sides set of two equations k like type in just y equals X. So one equation says y equals X instead of K. And another question Why equal sign of one plus one half k squared and find that point of intersection. So, uh, where do they cross? And I&#x27;m trusting Answer on this one that, as you type that in and find that point of intersection, um, you&#x27;ll see that that values about 0.9973 That&#x27;s a value for K that we want. I should put approximate symbols

Vinnu M · Answer

here. Go find the area under the craft. Why? I think all of us to works. We have toe in Tikrit. Here&#x27;s through two works. Introspect toe, do you? And here the limits of it given as from zero. Okay. And it&#x27;s given that there to sequel. Toe three. Where? Units. So first we have the value distant ducal. By doing that, he have he raised toe to x by two. Where the limit question ferial Okay, it is equal to three. Moving forward. We substitute limits for rich figured us toe. OK, miners he raised to zero. I do equal to three. From there, we can&#x27;t. Here&#x27;s toe to K minus one equals 26 here for here is to you. Okay, equals seven. Taking long onboard site. That&#x27;s for the dog You have. Okay. Cooling? No. Oh, so well, therefore you contrite kiss. Let&#x27;s lock off seven Divided by do

Julian Wong · Answer

this problem. We&#x27;re looking for the value of K and were given two equations y equals Keiko sex and y equals X squared. And this time, we actually have the area as well. Um, so we can certainly relate the area with an integral, but we need to know what the limits of integration are. So we&#x27;re gonna have to equate. Ah, Keiko sex and K X squared to find where the Intersect Because that&#x27;ll tell us the limit of integration. Um, so if I make co sex, Keiko, sex equals two K X squared case will cancel out. So we just have co sex equals X squared. We do need to use our calculated herself for X here, and that ends up being plus or minus zero point eight to s o. Take a look at that. The X values our paws oven negative. Same value, right? And if you actually sketched it, you&#x27;ll find that it is a symmetrical graph. So I&#x27;m actually going to consider the limits of integration from 2.8 to 4. That would give me half of the graph, and then I&#x27;m gonna multiply the overall area by two. So area is equal to two times the integral for 0 to 0.824 Rumor. That&#x27;s half of the area. The two in front deals with the second half. Um, if we graft it, Keiko sex. That&#x27;s the, uh, Kong Kong cave down sketch that will be the upper curves. Okay, co sex minus K X squared DX. And we also know that the area is equal to two. So too equals two and a girl from 0 to 0.824 and take out a common factor of kay here. So we&#x27;ll have co sex minus X squared DX uh, moving the two over. That&#x27;s one. The K is actually just a constant multiple, so we can pull that out of the integral. It&#x27;s okay. Integral from 0.0 0 to 0.8 to four of co sex minus X squared the ex. So one equals K and the anti derivative For this Ah, that will give us sign X minus one over three X cubed, evaluated from 0 to 0.8 to four. And ah, if you have a calculator on hand, putting 0.824 than putting zero in there, that will give us a value of 0.5 for seven, 3767835 Not rounding. This is not the final answer. And then isolating K that will give us an answer of one point. Eat to six nights.

Gregory Higby · Answer

eso This was really similar to the previous problem we&#x27;re looking at. X equals a squared. Why? Which can looks like this. Um and y equals zero like a zeros. This access X equals two right here. So we&#x27;re looking at this area, Um is my understanding based off of everything? Um, so, yeah, this functions X equals the square root of Why so if they tell you that X equals two Alice is rewrite the problem this way that X squared equals why we&#x27;ll just leave in terms of X and, uh, yeah, right here is X equals zero s. Oh, what I would do is find the integral from 0 to 2, because I&#x27;m putting things in terms of X now of the X squared function DX easy enough. That&#x27;s this in a rolls one third X cubed from 0 to 2. Which of you plug in to to Cuba&#x27;s eight, So it&#x27;s eight thirds minus zero. Okay, um, So then what we&#x27;re trying to do is the find the vertical line that could see area in half, so I don&#x27;t know exactly where I&#x27;m just going trough giant red line so that the area to the right is equal to the area till left. So basically, what I&#x27;m gonna do is just Well, I like that notion of multiplying by one half or dividing by two. Um, and what that&#x27;s gonna give me when I do that, um, eight divided by two is four. So what, we&#x27;re gonna do this, um, set up a new area problem, but I don&#x27;t know what this X value is. We&#x27;ll call it K. So we&#x27;re going to the integral from zero to k off the upper function is still X squared, DX. And so basically, it boils down to the same anti derivative from zero decay, and we&#x27;re gonna see when it gets us to four third. Well, if I plug in K here, one third k cubed minus zero. If I multiply both sides by three, that gets rid of both threes, and so K is gonna equal the cube root of four. And I would just stop right there. No need to try toe. Simplify that. It&#x27;ll

Patrick Vaughn · Answer

and this problem We have a region enclosed by the curves. Why squared equals K X and the line X equals 1/4 K. So if we plot that, what does that look like? So this is a parabola, like so And here we have, ah, vertical line. So we get this kind of crescent shape. OK, Onda, we want to rotate this around the line X equals one half, okay? Or assuming that K is strictly greater than zero, so are rotating about this. And we want to use Cylindrical Charles. So what does the shell look like? So if we take a typical shell Okay, so what is the height? The height is gonna be So, first, we need to convert this into some function of X so x. I&#x27;m sorry of why So why is going to be equal Thio plus or minus the square root of K X? So here does top curve here would be y equals route K X, and this curve down here would be why equals negative screwed K x. So when I find the height, the height is gonna be a squirt of K X minus negative squirt of K X which is equal to two square root of K X. Alright, so next we want to know. How far do we scan along? X? So what? Where is this intercept? Here. So where is 1/4 k equal Thio K X. So that&#x27;s gonna be when X equals 1/4 don&#x27;t you think? Oh, no, sir, That&#x27;s not very so we know X is equal to 1/4. So then we can just find we don&#x27;t even that we that there&#x27;s our grounds are bound to go from X equals. So X is going to go from zero thio, 1/4 k. Okay, so next we want to know what the radius is gonna be. So the radius when X is equal to zero, the radius is equal to one half k. And when X is equal to 1/4 K, the radius is equal to 1/4 K. Okay, so what we need to dio is our radius is going to be equal. Thio one half k minus x That&#x27;ll satisfy this. So when X equals zero when X equals 1/4 k. Okay, so now we&#x27;re ready to set up this lyrical shell. So we have V is equal to the integral from zero. So 1/4 k of two pi r, which is one half k minus x times the height which waas to root K x DX. Yeah, so that&#x27;s gonna be two pi out in front, integral from 0 to 1/4 K. Uh, so we can actually pull this to out as well. So this will be for so that would be one half. Actually, we just leave it on this. So that&#x27;s gonna be K route K x minus X two x route K X dx Thank you. That&#x27;s gonna be two pi and we can use power rule now. So this right, let&#x27;s tidy this up. So we have que route K route X or X The one half. Then here. What&#x27;s this? This is gonna be minus root K times X to the three house. But all right, so then, by powerful, we&#x27;re gonna have K Route K. Thanks times so x. So we&#x27;re gonna have we&#x27;re going to divide by three haps. We&#x27;ll have two thirds X to the three halves minus root K. In this time, we&#x27;re dividing by five hap. So we&#x27;ll have 2/5 x to the five halfs, and we&#x27;re evaluating this from 0 to 1/4 K. So what is that gonna be? So we&#x27;re gonna have two pi times. So we have K to the three halfs times. Two thirds. We&#x27;re gonna have, um, 1/4 K to the three halfs minus squirt of K times 2/5 times one force que the five halves. Okay, come. But okay, so we have K to the three halves times two thirds times. So what&#x27;s the square root of? 1/4 is gonna be one half. So one half cubed is gonna be 18 times K to the three halves minus K to the one half times to fifth. So we&#x27;re gonna have won over. So it&#x27;s four times 8. 32. So two of the fifth times k to the five haps. Okay, so if we multiply these case together, we get K to the six halves and k to the six halves. So that&#x27;s the same as K Cube. It&#x27;s okay. Cute. Okay, so we have two pi times, so to pai que cube times. So we have 1/4 times. Three, So four times three. Mostly don&#x27;t like that for now. minus one over five times 16. This is gonna make it easier todo when we combine these fractions. So what is what&#x27;s the common denominator? So we need three times five times Force Word. So this one is missing five and four. So that&#x27;s 20 minus. And what&#x27;s this one missing? Just 3/3. Time was three times five. Okay, so four square to 16. 16 times five is a 80 times three is 2. 30 to 40. Alright, so 20 minus three. So we have two pi k squared, cubed. All right, Uh, and we multiply this to to the denominator to 40. We&#x27;re gonna get out 1 20. So K K uh, pie. So what&#x27;s 29? 37 So 17 pi Okay, Cubes over 1. 20 e completes a problem.

Stephen Hobbs · Answer

to determine the area bounded between these two functions of this area and black here. We could go ahead and graph each of these and then notice, but the function to is on top of the blue one. And then we want to find our intersection points here. I&#x27;m here, and so you could use the graphing utility. But we&#x27;ll just go ahead and do this step by him. So we&#x27;ll set these two functions equal, and then we want to solve her isolate for our X here. Okay, so, um, we&#x27;re gonna do a couple steps at the same time here. We&#x27;re gonna, um, interchange. We&#x27;re gonna multiply both sides by two. And then we&#x27;re going to, uh, multiply that square root over and then square both sides. So it&#x27;s gonna be this on fourth and then add the X word over. Subtract the 1/4 over. So that will give us 3/4 is equal to X squared. But then again, this will be plus or minus X or another way to say that is does the same thing as negative. Through 3/2, founded on the left and bounded on the rate positive every three over to And so really just concentrating on our set up. Now we&#x27;re good to go. In terms of the integral, it&#x27;s gonna be the area from Negative three. Have Route three hands the positive Route three House of the top function to minus the bottom function one over Route one buying That&#x27;s expert DX. And although this looks, uh, a bit of a mess, the anti derivative of two is just to Lex, the anti derivative of the second part. There is inverse sign, and then we still have the same limits of integration. And so substituting does in and simplifying will end up with a final result of to brew three minus two. Hi, Third&#x27;s. And so that is the final answer for the area under the curve and black. There, that&#x27;s it.

Problem

Estimate the area of the region in the first quad…

Problem 31 Hard Difficulty

Estimate the value of $k(0<k<1)$ so that the region enclosed by $y=1 / \sqrt{1-x^{2}}, y=x, x=0,$ and $x=k$ has an
area of 1 square unit.

Answer

$$
k=(0.9973)
$$

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Howard Anton, Irl Bivens, Stephen Davis

Calculus Early Transcendentals

Grace H.

Heather Z.

Kayleah T.

Kristen K.

Integration Review - Intro

Definite vs Indefinite

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$$k=(0.9973)$$

Calculus Early Transcendentals

Grace H.

Heather Z.

Kayleah T.

Kristen K.

Integration Review - Intro

Definite vs Indefinite

Howard Anton, Irl Bivens, Stephen DavisCalculus Early Transcendentals

Grace H.

Heather Z.

Kayleah T.

Kristen K.

$$
k=(0.9973)
$$

Howard Anton, Irl Bivens, Stephen Davis

Calculus Early Transcendentals