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

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

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 6

Double Integrals

Partial Derivatives

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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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Find the Integral

Hello this video I'll show how to find it to go this expression here. So we have a dx which means we are integrating with respect to X. So if you look inside you have S and Y And one might be confused how trajectory y. So remember the most important part is to look at your D. X. So Dx tells us tells us that you're integrating with respect to X. So everything inside here which is not an ex were treated as a constant, meaning we're gonna treat why is constant. Okay so answering this so I'm just gonna write this uh function in another way. So I'll fill out our constant which is why and then inside we have left with X. Then I'm going to write this is my last one minutes. What you That's why the part 2? Oh this is the same as to the power of Yeah the X. Okay great. Now we need to integrate this. So here we use substitution. So I'm gonna make this part. Are you? So you were saying lit you? Yeah it's just you know yeah. Something that you Close to one when it gets to the cartoon. That's why the product you so if you want to find you this becomes minus two X. Dx. Okay now yeah we need to find face expression. We don't have it. We don't see it here. Deco so I want to do is I'm gonna write this form so that we can see clearly so we have minus how fear and they were from white. Take her off then here you want players Excellent. Part two. Why two part you bins. So this is off. And then yeah you see that I have -2 x. D. X. So this should be making racketeers when it is to confuse you. So if you take a look yeah This man is two x. We'll cancel these miners. Actually this managed to cancel this man is half so that we still have our expression we have above. But then what would you do if if we're done here particularly at this this is the same as what we have here and bring so we've got some shoot this do you? Okay great now progressing fed up now we're gonna make we're going into this substitution Menace House here and then substituting that we're going to have U. To the power of then now here do you? Great. No you take a beating. Mhm. So what's the integral of you 2.5? We know that this becomes you two or 3/2 all uh three of a cheap remember every plus C. Because this is an indefinite integral. And then whatever the left weeks now So to do a little bit of simplification so we are dividing by 3/2 Which is the same as not playing by 3/2. So what we can do here is we are not playing I actually just right here right so that you see better Soul -1 multiply by three. I'll play about two or three. We're gonna have minus 1/3 And then was two years old. White. And then remember how we made substitution of you? He said who was one man is excellent too. And has wiped out too. So we need to return that one minus X to the power to my nose. Why do the poor chip? And then this is the power three of 2 and then lastly live plus a constant. So this becomes the integral of our expression. We have the power Thank you for your time.

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