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Find the indicated volumes by double integration.The volume above the $x y$ -plane and under the surface $z=4-x^{2}-y^{2}$

Calculus 3

Chapter 29

Partial Derivatives and Double Integrals

Section 4

Double Integrals

Partial Derivatives

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So we've got another region here. Calculate its volume. We're going to draw our there regions got Weikel Czar X equals Y squared X equals four. So I got these points at four. Comma two and four. Common Negative. Right. And we're going to use a region too. So I'm going to integrate with respects two ex first and then with respects to Why so the bounds on why are the lowest values of why? To the highest values for okay now for X the lowest region the lowest values of X are bounded by this curve here, which is why I squared and above by this guy here, which is four. They got r inte grant And then we're doing d x d y. All right. So we got negative, too. Two of X plus y squared X cubed over three. And that's from four toe y squared. Why? So we're going again? From negative to to to So we play in four first for ex who got four plus why squared Times sixty four over three. Lina's target. Sorry. We need a apprentices here. If we plug in, why squared? Why squared? Plus why squared over three times Why not to the sixth here. D y okay. All right. Great. So now we want to kind of clean this up. It's going to be a messy one, but we'll clean it up. The best we can we've got is Allenby sixty one. Why? Squared over three nine s y to the eighth over three d y? Now we integrate in fact, what we can do before we integrate to make our job a little bit easier to say that this is an even function with respects to Why? So we pull out too, change our bounds. And, uh, no plugging in will be quite nicer. Let's say why. Plus sixty nine over sixty one over nine. And then, uh, why cubed? Minus. Why? To the ninth, nine times three is twenty seven from zero to. So we got eight plus sixty one over three times eight minus two to the ninth, older twenty seven. Okay, after all is said and done, when you combine these things, you will end up with this as your answer

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