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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 anarea of 1 square unit.

$$k=(0.9973)$$

Calculus 1 / AB

Calculus 2 / BC

Chapter 6

APPLICATIONS OF THE DEFINITE INTEGRAL IN GEOMETRY, SCIENCE, AND ENGINEERING

Section 1

Area Between Two Curves

Integrals

Integration

Applications of Integration

Area Between Curves

Volume

Arc Length and Surface Area

Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

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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's the deal. I equals X. We know that we're going from zero, and we're going to X equals K and the premises. We want the area between the two curves to equal one. So it'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're gonna fall into is what C and I dreamed of this. And a quick Google search, if you haven't learned it yet, is that is the inverse sign of X the and a review of Xia. I'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'm not gonna worry about. Science, Enver. Sign of zero assigns the Y coordinate. So right here. What's the radium measure? It's still zero, and I'm plugging in. Zero here will zero square to steal zero. Subtracting zero doesn'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'm trusting Answer on this one that, as you type that in and find that point of intersection, um, you'll see that that values about 0.9973 That's a value for K that we want. I should put approximate symbols

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