find-the-volume-generated-by-revolving-the-regions-bounded-by-the-given-curves-about-the-x-axis-u-27

Question

Find the volume generated by revolving the regions bounded by the given curves about the $x$ -axis. Use the indicated method in each case.
$$y=x^{2}+1, x=0, x=3, y=0 \quad 	ext { (disks) }$$

Gregory Higby · Accepted Answer

All right. So this probably have X squared plus one. Why it was X squared plus one. And they give you the bounds from Mexico, zero two, x equals three. Uh And they also give you Y equals zero. Which is kind of important because that&#x27;s telling you to take this region to revolve around the X axis. Um and if you didn&#x27;t know is the disk method, they go ahead and tell you because you have these perfect discs that go across. So the formula for that is pi times the function square in terms of X. Um our bounds are zero and three like they told us and it&#x27;s X squared plus one. So what I would do is foil that out. So we&#x27;re looking at X to the fourth plus two, X squared plus one dx. So when you evaluate this integral, you just have to add one to the exponent and then divide by your new export. So this becomes 2/3 x cubed Less x going from 0 to 3. So at this point I&#x27;d probably use a calculator. Um Just to plug in 3 to the 5th power. It&#x27;s 243/5. Uh now this next one, the one of the three&#x27;s cancel with this three. So just three squared is nine times two is 18. But you can double check that arithmetic and then if you plug in zero a bunch of times, just get a bunch of zeros. So as you type this in 2043 50 plus 21 change it to a fraction. I got an answer of 348 tests and then just make sure you multiply by pi Okay.

Adnan Gill · Answer

So we have this graph here we have. Why is equal to extremely powerful E which means that X is equal to lie to the power wanderers. And then we have our line X is equal to two, which means at this point would have coordinates to hate. So the area is the area bounded here. So in this area we take this elemental piece here than our inner radius or smaller radius. R is equal to y to the power one or three and the bigger radius radius R is equal to two. So we have ah V is equal to the grill from 0 to 8 high times R squared minus R squared d y. So we can put the pie in front of Tina Grill and we can see it. The volume is pi times integral from 0 to 8 and R squared is two to the power to minus y to the power 1/3 square. Why? So he&#x27;s equal to high times the Negril from zero to the 84 minus y to the power to earthy Do you? Why? So we can now actually apply the rules here so musical too high times for why, minus y to the power. 5/3 times. 3/5. Limits of integration from 0 to 8. So he&#x27;s equal to high time. Four times eight minus three times eight to the power 5/3, divided by five minus on the nor limit will just be zero minus you. So we have ah V is equal to pi times 30 to minus 96 or five O. R. V is equal to 64 hi over five cubic units.

Russell Arnold · Answer

so because we&#x27;re integrating along the Y axis were going to want on exit in terms of why, instead of lie in terms of X so we have on that why equals X to the 2/3 We can re write that as X equals X equals y to the three halves. And this is just the dismount that no washers. So an integral 02 of pi times x squared, which is just going to be very conveniently Why cube deep? Why? And so this is just going to be hi over three times too cute just ate.

Bobby Barnes · Answer

we want to find the volume of the solar generated. When we revolve. The region X is equal to over a square of wide plus one exit with zero y z zero and wives three around the Why access. So verses go ahead and write down what formula we want to use for this. So since we&#x27;re rotating about by access, we would want to use some interval from A to B, which will be our pounds of integration, some radius function r squared. And in this case, they&#x27;re gonna want in terms of why we&#x27;ll need to multiply this by high and will integrate with respect to why let&#x27;s go ahead and just kind of color in the region that they said we&#x27;re interested in. So it said from why is equal to zero by three. Well, that would be here here as well as X is equal to zero, which is here. We&#x27;re going to take this region and take some point on it and rotated around. So it&#x27;s going to be this here, and we rotate around B y access like that so that green line that have grown there should be our radius. And in this case, That would just be the distance. We are along the X axis and we&#x27;re told that distance is just too over why plus one and all of that square with So we have our function are in terms of why and now we can just go ahead and plug that in for first up. One of the things I&#x27;m gonna do remember, we can pull that hi out in front of the integration and then we&#x27;ll want to square two over this. Where the y plus one. So in the numerator we&#x27;d end up with or an end of the non there. We just have wide plus one when we still have our d. Why Over here now we need to figure out what our balance of integration are going to be. And lucky for us, we don&#x27;t really have to do much because they already tell us that. It just needs to be from why isn&#x27;t with 02 three So about zero here and three there. Now, what I&#x27;m going to do for this to integrate it is using U substitution, so weaken right on the side. But you equal live plus one, because this kind of looks similar to the integral for natural log. So if we make this substitution here, we can go ahead and take the derivative and get the differential. D&#x27;you is equal to Dubai and then we can just start plugging things in. So first time to pull before out front. So I get for pie here. I can go ahead and replace why, plus one with you. So one over you and d y just becomes now. The last thing we need to do is figure out what our balance and expression are going to be. So we have when why is equal to zero? Well, plug it zero into that will give us you isn&#x27;t one. And when? Why is it it&#x27;s three for getting into that equation? Can we get you? Is four so we can go ahead and put one or here and now we can integrate. So we get four pi won over you into role bins, Natural love of you and then we want to evaluate this from one so we&#x27;ll have four high natural log of four. My next natural log of one and the natural log of one is zero. So we&#x27;re just left with or pie natural log of four for the volume. When we rotate around the Y axis

Russell Arnold · Answer

So for this problem, Um, hey equals pi r squared, which is just pie. Excuse squared Pi x six and the is the integral From 0 to 2 of a the bounds are given George too imported by out here. It is easier to look at on DH. We get it. Pi over seven times. 120 eights tooth, seventh power.

Bobby Barnes · Answer

We want to find the volume of the solid generated by revolving the region&#x27;s. Why is even two X squared plus one? And why is it X plus one about the ex access? So first, we should decide on what formula we want to use. And any time we&#x27;re trying to rotate areas between two curves around the ex access, the formal to use is into goal from 80 of some outer radius squared minus. So inter radius squared and we integrate with respect Next. Then we need to multiply by, so to figure out what our rady I should be. So since we&#x27;re rotating about the ex access, we start from here and then go up. So the first time we hit will be art in a radius. So little r of X is going to be expected plus one. And then outer radius is going to be our second line. We hit so capital R of X is X plus. So first, let&#x27;s just go ahead and plug these in cilia pie over here and believe off the bounce of integration. Right now, we could figure those out in a minute, so I&#x27;m just going right X plus three squared minus X squared, plus one squared. Now the thing we&#x27;re gonna need to do to find our balance of integration is figure out what these two points are. Somewhere these lines intersect each other, so let&#x27;s go ahead and see if we can figure those out. Well, to do that, we we just need to set the lines equal to each other. So refined. Have X Plus three equal to X squared plus one, and we can move everything over to one side. So Mr trucked X and three to get zero as he could to expert minus X minus two. And luckily for us, this is something that factor&#x27;s pretty easily I&#x27;ll get X minus two X plus one. And so, using the zero product property that tells us X is equal to as well as X is equal to negative one. So those will be our bounds of integration. Now we just need to do a little bit of algebra here to claim for something. And then we can start integrating so equals pi into girl from negative wonder, too. So boiling that out, we get X squared plus six x plus nine minus X to the board waas two X squared plus one of RDX out here. Ah, now, just general. But more algebra, so x squared and expert can go together. So the expert minus two X we end up with a negative x squared. Um, there is no other ex term, so we can just go ahead and bring that down. Plus expects. We have nine minus one. So that gives us plus eight. And then we only have the extra four term pierce. We have negative X e X. Now we could go ahead and use the power for integration for all these two plug or toe, find out what this and I derivatives should be. So we end up with negative X squared negative X to the third power. Then we want to divide by the new power plus six X now to the second power. We&#x27;ll divide by the new power plus eight X and then minus X to the fifth. And then divide by the new power. And then we&#x27;ll go ahead and value. It was from negative 1 to 2. All right, now we can go out and play to win. So hugging two into the first term will end up with negative earns. And actually, before we do that, let&#x27;s go ahead and simple. But it&#x27;s a little bit so the six over to we can actually simplify down to just three, and it&#x27;s looking simplify and then two squared is four times three. So we end up with plus 12 will gain to into their we end up with plus 16 and then plugging, too in two extra if we end up with negative 32 over five. Then we subtract off pi times when we plug negative one into here. So X cubes for negative one huge is negative one, so that first term ends up becoming 1/3 than negative. One squared is one. So end up with plus three negative one times eight is negative. Eight and then negative one to the fifth, this negative one. And so that becomes one. No, you go ahead with the algebra Phyllis, and when you do, you should end up with 117 times pi. So the volume of area when rotating this Balan did region here around the X axis should be 117. It&#x27;s times pi

Problem

Find the volume generated by revolving the region…

Problem 13 Easy Difficulty

Find the volume generated by revolving the regions bounded by the given curves about the $x$ -axis. Use the indicated method in each case.
$$y=x^{2}+1, x=0, x=3, y=0 \quad \text { (disks) }$$

Answer

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