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Evaluate the given integral.$$\int x y \sqrt{1-x^{2}-y^{2}} d y$$

$$\frac{x\left(x^{2}+y^{2}-1\right) \sqrt{1-x^{2}-y^{2}}}{3}+c(x)$$

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 6

Double Integrals

Partial Derivatives

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Lectures

12:15

In calculus, partial derivatives are derivatives of a function with respect to one or more of its arguments, where the other arguments are treated as constants. Partial derivatives contrast with total derivatives, which are derivatives of the total function with respect to all of its arguments.

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Evaluate the given integra…

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Evaluate the integral.…

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Evaluate the integrals.

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Evaluate each integral.

Evaluate the given double …

no this video I'll show you how to find me to go with this expression again. So we have why here which is an important point out of. So inside here you see that we also have X and Y. And one might be confused. So like I said Dy is something to take not off. So do I mean is that we are integrating with respect to Y. So everything inside this expression of function rather which is not why we treated as a constant meaning not only one is a constant but also X as a constant X. Where there's a constant. So that's how we're gonna treat this function here. So answering this, I'm just gonna write uh this is the man the way we should help us to integrate so affected out the constant the X. They gonna be left with wine then you you it's expected. Yeah apologize some here. Thanks to what you minus Y to the power to mm all this in at a place of the screw looking right to the behalf and you know you have our do I? So looking at this what we can use here is substitution. So I'll make this field. So here we are saying I'm great thank you is equal to one minutes X. To the two minus Y. To the power to great. And if we want Dy sorry do you do I So we are differentiating with respect to why blankets use you in minus to I. Yeah then if we only want the Y on the left hand side this becomes mass to I do I I saw take note of this we're gonna use it. We do our substitution soul looking at our expression I wrote here. We can also write it in form. So generated this. Fun Yeah. Rate of excise constant. And then we have one Extra Day or two. Yes. Why not you this system. Uh huh Dearly. About why. And our do I? So if you take a look here when we introduced our substitution we had We have -2. Y. Do you like? But then you have why do I so you can just make a constitution right away. So what we need to do is we need to make this function appropriate for substitution. So we can introduce manners half outside here do we have to walk constant X. Then they gave you expression inside you. Mhm. Now that we've introduced now that we've introduced minus half. This becomes minister to Y. Do I brother opened the brother here. So remember this pocket it shows that we are multiplying so it's not subtraction. So the white. So now if you take a look at this, this especially heavy now mixes with how we define our substitution. So now implementing this substitution here for question as it is minus half X. And then inside remember we made the see you. So when I have viewed the power of then now you can replace this with you so you have to do yet and then integrating this. Yes off X. And what's the integral of F. U. To the power half? We know that this becomes huge, Power three or two Over 3/2. And remember we have a plus constant because it's an indefinite integral agreed. Now we're almost there now here we are dividing by 3/2. Which is the same as not playing by 2/3. So minus half for playback to over three. This becomes mess one more three. And then we have X. Now remember how we substituted you? So isn't uh who said let's you is the pursuit one minute extra support you? And that's why I put you So waiting back. Yeah, while an extra part two That's white or two please, this man. Mhm. And that's why the part two, remember this is you have a tube and not stealing here. Lost. So this becomes the integral the expression you hear about thank you for your time.

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