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Solve the given problems by integration. Find the area of a quarter circle of radius 2 by integrating $\int_{0}^{2} \sqrt{4-x^{2}} d x$

Calculus 1 / AB

Chapter 28

Methods of Integration

Section 8

Integration by Trigonometric Substitution

Integrals

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All right, so to the penis of the integral from 0 to 2 of route four minus X. Squared Dx. So uh from the start you can draw the triangle. I'm going to draw the triangle, find the value of X. And data, plug it in a big vision. So I'm right during the right angle triangle. Uh huh. There are 2000 mobile here. And this would be two and X. And then paste with the four minus X square. Now from this strangle you can can't do that sort of four minus X square. Mhm two and secret first theater. And that gives you a brutal four minus X square as too costly to and from the triangle uh you can get scientist to as popping up on my partners ex by two so X by two people just signed later. And so you get the X. As the differentiation of two centimeter which gives you too costly to the theater. And now uh can even get fitter in terms of X. That's your new does. You've got to sign and worse explain to so you got to stand and was expected to Now you have all your variables on the right hand side. Now you're just talking to the formula um to solve this question I'm gonna ignore the upper hand and the Lohan limit and just solve it as an indefinite integral and at the end of between the weapon and limits. So uh it's theirs. You know this has to because data the teacher into the past data as dX is to pass to defeat to and the root of formats experts to past data and no you bring the two outside. I'm not gonna matter that these two you'll see why um something you get two cross square Yeah defeat to if you remember on electric no metric identities as to costs where data as you could do minus one change of course to to to so you can write this as two. Integration of course to peter last one peter Sure. And then you bring the two inside on the diet and you get integration of to cost to theater fast too the theater. And you can integrate this by getting to sign to theater less to theater. So you have integrated your interview and now you need to in terms of X so signed with the attack and your willingness to sign the mosquito another Katrina metric identities plus two. We got to to as san universe expect to so you throw that in so you have that now and after this. You need to uh simplify this a bit further. So you've got a scientific to as Xy two soviet and so stand united text by two. So you went to expect to and you got to ask you to as car route for my thanks very upon to first to sign investors expect to so you can recognize this as simplify it as excellent for lines X square overtones. And now you have it in terms of X. I'm gonna bring in the defense back again are getting this done as to sign. And worse text by two from 2 to 0. And you plug you in the upper bound in the first, uh, front side of the equation got zero and over here many plug in to enter the Simon worse you get by and when you plug in zero you get zero. So find 10 She'll give you buy. So your friends answer is bye. So I hope you guys understood that. And thanks for watching.

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