Find the volume under the given surface $z=f(x, y)$ and above the rectangle with the given boundaries.

$$z=y x \sqrt{x^{2}+y^{2}} ; \quad 0 \leq x \leq 4,0 \leq y \leq 1$$

$\frac{1}{15}\left(17^{5 / 2}-1025\right)$ cubic units.

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Okay, We want another volume but we'LL just go ahead and set up the double integral. Just taking the the volume underneath the curve of this function. Get in here and doesn't really matter which variable we put first actually here symmetric. So the function is symmetric I need So it really doesn't matter s o get why X in square it of x squared plus y squared. Let's see a foot. Why first, Why on the insides and steal white yaks. Okay, so we want an anti derivative with respect to why this looks like a substitution problem. So we have Ah you this x squared plus y squared well, different shape with respect to why don't you just use Teo? I why or in other words, why do I And on this one one year Okay, so I'm here for now. We have a factor of one half fromthe substitution here and then are you limits are now. Okay, so we're plugging and why is your one so that's going to be X squared toe one plus x squared of Okay, this is going to be well. We have this extra factor of X, which is constant on the inner integral soul factor that out there and then we had Why d why was absorbed in the substitution? And then we're just left with root you Do you d x So again this is We've done this a lot Neater. The one half anti derivative is, you know, the three house divided by three house. And so if we factor all that won over three house we have one half divided by three house, which is one third in a girl's here for you have an ex and now way have you to the three halves. And we're just evaluating at these two points. So that's going to be one plus x squared to the three halves, minus x squared to the house. Jax. Okay. And so what can we do here? Well, in both cases, this is not going to be too bad, too. Taking it a derivative. Because if if we just look at that inner part, if we take the derivative or just going to get to eggs on, then we'LL have the DX, so we'LL get another factor of one half and the substitution service will end up being one sixth and then it's just going to be This is just going to be like you to the three halves. So then we're going to have to get add one to expense will get this to the five halves of her five halves and this to the five house over five pounds. So that's dividing by five halves are multiplying, but two fifths. And then the ex gets absorbed in the substitution. The X t x, we have the factor of a half, and then we have one plus x squared five halves minus x squared to the five halves evaluated from zero four. Okay, so this is wants to her third. So one fifteen and then we plug in for I won't get to see. What is that? Seventeen to the five house when you're playing for over here only plugging for he will get. Okay, So that's for to the fifth, which is two to the tenants, which is one of those brain of things I have memorized. That's ten twenty four. Okay, so you get minus ten. Twenty four. Now we have to evaluate it. Zero, this is going to be zero, and this is going to be one machinist. Attract one. So this is just one fifteenth, seventeen. The fight halves minus ten. Twenty five. That

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