find-the-indicated-volumes-by-double-integration-the-volume-above-the-x-y-plane-and-under-the-surf-2

Question

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

Alan Ghazarians · Accepted Answer

So we&#x27;ve got another region here. Calculate its volume. We&#x27;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&#x27;re going to use a region too. So I&#x27;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&#x27;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&#x27;s from four toe y squared. Why? So we&#x27;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&#x27;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&#x27;s going to be a messy one, but we&#x27;ll clean it up. The best we can we&#x27;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&#x27;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

Alan Ghazarians · Answer

Let&#x27;s draw X Y plain. We&#x27;ve got the circle from negative to Teo too. But know that we&#x27;re only looking at the first oxidant, right? The first talk tint Is this guy here? Because this is the first often so we see that our X is going to rain from zero to two. And our why is going to rain from zero to the square root of four minus x squared. That&#x27;s just by solving this guy for why good and our double into girl. So let&#x27;s get done. Black are double intro going from zero to two from zero to square root of four minus x squared. And, um, we&#x27;ve got the surface above as equals X y and below x minus R x plus y. So this is the surface that&#x27;s above or on top and below like that. So what we&#x27;re doing Do I tx So from zero to two, we&#x27;ve got X Y and excuse me, we don&#x27;t Yes, we have X y plus why squared over two minus X y squared over two from zero to the square root of four minus x squared DX. Okay, uh, now we just plug in amusing FTC here plus one half of four minus X squared, minus tax over to for minus X squared, Dex. All right, so this is just a very standard and to go problem, this guy here we&#x27;re gonna have to use use of. So when we do use of, we&#x27;re going get zero. You get a negative out, we&#x27;re going to get a four to zero to the square root of you. And you&#x27;ve also got a one half out here to you. Plus the integral from zero to two. Two minus x squared over to minus two X plus X cubed over too DX. Okay, So what is our next step? Well, we need to integrate this, right? So integrating this is not too difficult. We see that was that one half. We&#x27;ve got two thirds can four to the three. Oh, her too. Plus two x minus, X cubed over six minus X squared plus X to the fourth overrate from zero to two. And also done. We&#x27;LL get thirty two over three

Uma Kumari · Answer

hell and this problem we have given the equation Z is equal to four minus two y Well, these equals to zero X s equals toe by toe. The bubble fall on X is equals to one. We are asked to find an integral equal to the volume of a solid bounded by the given surfaces. First of all, let&#x27;s share the graph. This craft we have to find the idea of this region means we have to integrate this region so we can write double integral function is four minus two way dx dy. By now, the limits for the eggs will be Why to the power before and one on the limits off the vile will be minus 1 to 1. Now we will integrate this function with respect to X vacate integral with the limit minus 1 to 1 four minus two by into eggs with the limits Bye to the bubble 4 to 1 de by. By putting the upper and lower limits, we get integral with the limit minus 1 to 1. All my news four wise to the Pablo fall minus two way plus two y to the power of five The way now we were going to integrate this function with respect to why we get four by minus four into by to the power five upon five minus y to the power toe minus Phi Tau. The cover six upon three with the limit minus 1 to 1. By putting the upper in local limits speak. It tried to do by five. This is the final answer.

Carlos Pinilla · Answer

just before you with the region, uh, region. But he said to have these para veloute c is equal to five minus x 906 Plus why squared family was good. Minus these uh, the, um got her name between these Won&#x27;t see she won&#x27;t before for exist square Plus why squared? So these air to problem, it&#x27;s on. Duh. When you look here word a three dimensional space Why six i z these this probably lawyer should look something like that. Sort of like these from the these one is gonna be So what are you here is gonna be five So something when I hide five on being well known so even obtain like a region like that I should be symmetric with respect to rotations Our Vera So So how are two bounds for these? You will be between those universal bubble bad more Knicks. It&#x27;s why squared I really there is a happy above all these are below off these one life mine six were minus y squared when we can use, uh, cylindrical corner of the sun You see metre logistical mix Why c max investment with those soon. So so you can describe it through cylindrical coordinates because X squared plus y squared Elise R squared. So we have something in peace. So see how sweeting or art square y minus r squared so that allow those are the bones for C onto few year old. After which value goes, Are we help here? Speaker drawing over there so far I mean respect Z So you love these curve for our square is going to see with the Kirby&#x27;s five. My sir squares of a d. R. Will go from zero up to this point. It&#x27;s called These one are not so for these point should have that those two equations are equal. So we have four are square sequel to five minus are square r equals who are not so that we removed that they&#x27;re gonna get miners because supply supply. I didn&#x27;t that $5 Question five. So we get these desirous positive. The only solution is Article one are positive. So this point will be one. So I would go from 0 to 1. Uh, well goes over their own. You can rotate so the region can be seen us like you have these. You did it well. There the whole term So that said I would go from syrup to to buy a holder. So the volume the bullet was gonna be Why so for Syrian those bound So for within for our square arm five star squared movie See? No, no, the volume on this are are dizzy Would be that then for our she will from zero off the one on Thera from sear off to Dubai. Really? The seems, uh useful you before things independent of better. We can&#x27;t theory within a separately. So they drove from 0 to 2 pi incident. That would be just Syria. I really didn&#x27;t think to buy zero todo by. We&#x27;re actually really that internal in these illegal to to apply times from zero the one from four Art Square five minus are square Are this you are huh? So all we have here here&#x27;s being geo off this year The drill of the seat There is this constant with respect to see we just see so that evaluated meeting by my start squared before our square This is a beautiful tune. No, by minus R squared my hose that minus four. Our squares of these needs to be said it&#x27;s up to my minus five to these. Or is that so? That these were turning to brown you by it was the interval from zero up to warn Fi minus five part square. We are So we&#x27;re here. We have being drills. Well, the trouble So carrying to buy the journal five is just five are on the drum. Our square is our Q thirds by the bar rules another minus five nursing people to these evaluated one since you which is not me fool to buy them some five things one minus five times one cube. First we got minus Sear them by I&#x27;m seeing a cube thirds thes part zero His ear So with people too is boiling with people to find James five minus five hertz we can write that began. In fact, with five two times fly by one minus 1/3. Eso won with this one thirties will do so long as you will go any thirds. So is it even true? 3/3 miners, 1/3. The lightness like this one bring us warm. Third, to be able to sleep minus 1/3. These things, if I can find out there, is that So this is just too. Then they&#x27;re still by That&#x27;s these will be ableto to anybody. Good, like three. So we&#x27;re this This will legal dio to the volume of these region. Well, about begin over there. So William, do any by over three.

Uma Kumari · Answer

hell in this problem, we have given the portion of operable er disease equals two X squared plus y square below the planes is equals to four. We are asked to find the surface area of the given surface. We have these equals two F or X coma vile which is equals two x squared plus y square, then the normal vector and would be equals. True four pecks coma A four by Goma minus one which is equal to two weeks common to where comma minus one. Then the mold and her big was toe on the road four X squared plus four by square Last one. Now we have double Integral s. Yes, which is equals to double integral are more than the here s Is this office off? A caution of comparable weight on art is the region enclosed by the intersection of parable oil. Now, by using the polar coordinates, we have X squared plus y square is equal to one square under Route four x squared plus four y square would be equal plus one would be equals toe under route for our square plus one on they will be equals two R D r d theta So we get the double Integral are on the road four X square plus full by square plus one day would be equals toe double integral are in two under room. Who are our square plus one? They are deep de Tita. There are ranges from zero to So we get the limit. 02 on Tita ranges from zero to pan. Now we will integrate the in the limit with respect the wall on. After putting the upper and lower limits, we get one upon 12 into integral with the limit 0 to 2 pi 17 to the power three by two minus one day, Tita. Further, by integrating dysfunction with respect to theta, we get the final answer which is equals to buy upon sets into 17 to the power three upon to minus one.

Problem

Find the indicated volumes by double integration.…

Problem 21 Easy Difficulty

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

Answer

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Allyn J. Washington, Richard S. EvansBasic Technical Mathematics with Calculus

Lily A.

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