find-the-indicated-volumes-by-integration-describe-a-region-that-is-revolved-about-the-y-axis-to-g-2

Question

Find the indicated volumes by integration.
Describe a region that is revolved about the $y$ -axis to generate a volume found by the integral in Exercise 27.

Gregory Higby · Accepted Answer

what you could do to revolve something over the y axis is simply change it to be in terms of why 1 2 of why cubed. Dy So when you examine that, uh you saw the pie in front so it&#x27;s still the disk method, but instead of being X equals one, you&#x27;ll have y equals one and y equals to a problematic. You&#x27;ll also need the y axis which is X equals zero that we are revolved around and then one more thing to think about it is the disk method is going to be high times the inner role of something squared. So just think about it for a second that to get this to be squared, um Something squared to be cubed in a power to empower you multiply the exponents. This has to be 3/2. So the line that we&#x27;re going to revolve or the curve is X equals y to the three halfs power and now it satisfies all of these conditions and you could examine what it looks like like was one like was too um you know, you have some graphs, I don&#x27;t know exactly how it&#x27;s going to be curved but then it&#x27;s going to rotate this way and you&#x27;re going to get the same area as the previous problem. Same volume, I should say

Foster Wisusik · Answer

All right, This question wants us to revolve this region around the y axis. So first, let&#x27;s look at the slices we&#x27;re gonna have to make to complete this. So looks like we have two different behaviors going on throughout this region. So between zero and two, as you see, all slices have the same with the distance from the from the axis of revolution remains constant. So where? Why in the range zero and two, the radius just equals two. And you can see that we&#x27;re only dealing with one radius here because this is touching the access of revolution. Then after you get past two, you see that our slice depends on the blue curve also. So, using our rule of top minus bottom done from 2 to 6, the radius is two minus our function of why. And in this case, since why equals X squared plus one, we can solve for insult for acts, which means that X equals square root. The positive one of why minus two, this should be a two. All right, right. And that&#x27;s our f of y. So now let&#x27;s find the volume. Since we have two different behaviors will have two different girls, so v equals pi times the integral from 0 to 2 of the radius squared, which is to plus high times theater girl from 2 to 6 of our second radius to minus square root of why minus two quantity squared. Do you? Why and plugging this into a calculator Using the F in integrate function, we find that this turns out to be 32 over three times pi.

Shreya Kelly · Answer

for this problem. We are trying to find the volume of the region bounded by the coordinate access. Why is it called Ln X and Y is equal to one So which is nothing but the strait region meditating this red region about the X axis. Okay, because this region here, which I&#x27;ll show in green, is not continues its piece wise convenience. We need to split this region here at this point and the left, we have one set of integration to the right. We have another set off integration. Let&#x27;s just blue color here. So the particular point between this point and this point which is zero on this point, is one Between these two points at a particular X, the height is just one. So height is nothing but the radius. If you rotated of our gigs access. So this radius is going to be one. Say it is going to be by our script which is playing one square, which is nothing but pie. I think stressor I one square is 1/4 quite and to find the volume, we need to indicate the area with the land element, which is DX and experience from 0 to 1 a pen. Similarly, for the region to write off that green line discussion by Why is it called on the next here? Delete the blue line, the white confusion. So it&#x27;s cushion at the bottom by that line and on the top, that is, why is it called the one? So if they locate that region of our takes axis, it&#x27;s going to be this little circle and a small circle. So it&#x27;s going to be a donut. It&#x27;s only going to be the shaded region between the circles and the area of the shaded region is by capital our spirit by my small. Our script Capital R Squared is one script. Small are square. Is Olympics all square, So that&#x27;s going to be the area to find the wall you did he expect and apply the limits of interrogation. So at this point is one on this point. Here is going to eat, eat so X rays from one DUI. The reason that&#x27;s E is that we have icicle toe one here and bicycle Ln X here, so one encircled or Ln X gives excess, according so from excessive for the one be. This is the limits off indignation on this sustained Tikolo. So for this one, it is high. BX is nothing but X zero on one. So one pipe, it&#x27;s nothing. Mark Pipe And he had This is by one minus and the next hole square when we are taking the in people here. So that is pi minus pi Alan ICS once there began also, I did that still celebrating DeGross So this is high eggs from one to eat. And this one Let&#x27;s triangulation by parts So pie is common. So let&#x27;s take the by outside and we have Ln x oh scream. I&#x27;ll pay it up This piece here Namfrel Next whole script. Let&#x27;s do this. Let&#x27;s take the and the next full square. Do the outside. Let&#x27;s keep the one inside Minus the data way to appellate explores varies to a Lennix Mavericks They dig a loved one Is X holding different? So the Sylvie Ellen X whole square mile deployed with extendable upon is X minus. Needs to get cancelled and weapon up to indigo Ln X So we cannot play indication by parks again on the next hole spread my Mr Integral off clinics will take that the next outside Keep the one inside mine instead of weight off the nexus one over X and the lump on his ex. Well, in pickle, the Sylvia Exelon Knicks Well squared minus two. Thanks. And the next minus minus is plus that&#x27;s too We he canceled ex annex in developed on ethics So plus two x So this is the answer for this part. Modify the pipe. So let&#x27;s on a pie here a pie here on a pike you from want to eat? So for the 1st 1 it is high e l any is one Swan square is one minus So high e replied with one plus two e pie. I used to get cancer and for the bottom part, Ellen on the scene island on the Syria It&#x27;s only the third Uncle Bill. So that&#x27;s going to me. Do I so high? E minus two high is going to be lending. I love this one. That&#x27;s right here. The 1st 1 it is Hi e minus pi, but this part it is Hi e minus two by But there is an 80 of science so minus i e livestock high needs to get cancelled. Don&#x27;t by my most pie in spite. So this party is high class parties pie. So final answer is going to be pie less pipe, which is too high. Thank you for watching.

Taylor Shimono · Answer

So for this problem, we are going to be finding the surface area using a rotation brought the y axis. So that means that the formula we&#x27;re going to be using is the interval from CD of two pi times G A Y times square root of one plus G prime, a wife squared D Y and all the information that we&#x27;re given in this problem is in terms of X. So our first step is going to be converting it to be in terms of why so I&#x27;ll start with our boundaries are lower limit is going to be X is equal to one. So if we plug that in the cube root of one is gonna tell us that Y is equal to one and for our upper limit, Queudrue of 27 is going to equal three. Um and then also we can see that were first given f of X. So take your work this into a Jew y function and we&#x27;re just going to cue both sides. So G ive y is equal to y cute. And lastly, we want to find g problem boy, which we know is going to be equal to three y squared, Um, and to make this a little bit easier to plug into our equation, I&#x27;m going to square this ahead of time. Oh, this is just personal preference. So ji from Y squared is equal to you nine y to the fourth power. And now we have all of our information we need to plug into our formula. I&#x27;m sore surface area is going to be equal to the integral from 1 to 3. Our new bounds that we calculated of two parts times g of why What we said is three y squared. Oh, and then we&#x27;re going to multiply by the square root of one plus nine y to the fourth power do I and s So for the next part of our problem, we&#x27;re going to go ahead and, um, calculate T ah, you substitution here performing u substitution. I mean, and so I want to choose you to be equal to, um one plus nine y to the fourth Power. And I made a mistake in my setting up of my interval. I put three y squared. Um, I put three y squared. We&#x27;re going to move this over here, um, as my a term inside or as my g of why but actually my job I should have been just y cubed. I was so going back to my u substitution, I chose you to be equal to the term inside of my radical. So that means that Dio is going to be equal to 36. Why cubed, do you? Why and when we substitute this back into your equation? Uh, this is going to become the integral from 1 to 3 of a pie over 18 a times the square root or actions going to write times you to the 1/2 power, do you? And so in the end, right, we know that we&#x27;re going to leave our coefficient the same multiplied by the reciprocal of one more than our explain it. So the reciprocal of three house will be times 2/3 times you, which we said was one plus nine y to the fourth power raised the three house power, and this is all evaluated from 1 to 3. And so when we plug in all of our terms and simplify, we find that our surface area is going to be equal to pi over 27. Sometimes in parentheses 730 times square root of 730 minus 10 times the square root of 10. And this is in square units.

J Hardin · Answer

this problem is from chapter seven section to problem number sixty one in the book Calculus Early Transcendental. Lt&#x27;s a condition by James Door here we like to find the volume obtained by rotating the region bounded by the curves. Why equal sign X y equals zero xon between pie over to and pie. And we like to rotate this about the x axis over here on the right, we have a graph of y equals sine X for X between pyro too and pie. This is in blue. We also have in red y equals zero and then we have the bounds. X equals pi over to this is in green. This is the line vertical line X equals power too. And then our upper bound for exes over here at pie so we could see that the region bounded by all of these curves Is the area inside here? This object will seek a fin So it&#x27;s commented, called Eurasian buy Some letter was calling our and moreover, rotating this are this region about the X axis. So some notation that used might see here is something like that to let us know that we&#x27;re rotating this thing around the X axis so we could go out and actually draw what the solid would look like after the revolution after the rotation. So after we do so we get an object that looks like this, and then we can go ahead and make it look like a solid. So this thing with Wade you right here this our disk This is a cross section and we see that it&#x27;s a washer, so let eh be the washing area. And we know that our formula for the volume the is that integral of the area of the washer. So it will be the inner world, eh? So we need to be more specific here and our problem. The is a washer. So the area will be pi r square. That&#x27;s our into grand. And this problem we obtained the volume when this washer moves in the extraction. So we should be integrating with respect to X. And in doing so, we should use the bounds pirates who and pie for limits of integration. Also, we need to find our which is the radius of the washer. So are the washer radius. So let&#x27;s go to our picture here. Let this be our starting point. So we see that this is the center of the washer, the center of the disk. And if we move straight up, that&#x27;s a radius right there. And this is just some why value. So our equals Why we don&#x27;t want to use why in the general, Because we&#x27;re in a gated with respect eggs. So we use the fact that here this why, Val, you were on the blue curve. So why is given by signing books? It would be better to use this because now we&#x27;re in terms of X. So are integral becomes less claw that pie. It&#x27;s a constant power to the pie. And then we have signed squared of X. To evaluate this inaugural it&#x27;LL be best to use an identity a triggered a metric identity for sign square. We can write this as Science Square is one minus coastline of two x, all divided by two. And since two was just the constant, we could also pull that to a outside of the integral. And we can evaluate each of these in a girl&#x27;s and a girl. One is X, and the integral of coastline of two weeks is signed two eggs over too. Where am I help you to use a U substitution here. If you can&#x27;t see this for this integral, you could try and use up. U equals two X. Can I look and let&#x27;s not forget our end points piracy Tobi. So it&#x27;s good and plug these in for X if we plug in pie first. So let&#x27;s do that. So we have pie minus sign of two pi over too. Then we plug in pi over two for X minus. Sign a pie over to you. Separate this from our scratch work on the side. So sign of too high from the unit circle. We know that zero sign of pies also zero. So we&#x27;re left over with hi over too. And then pie minus pi over too. It&#x27;s also a pie over too. And so we got our final answer for the volume. Pi squared over four and there&#x27;s our answer

Russell Arnold · Answer

So here we use the washer method. So we have a VAX equals big R squared pi times big R squared minus little r squared which in our particular case, equals pi times four minus nine for its X squared. So then the volume is the integral from 0 to 3 of this right here, So fine. Plans for money assigned fourths x squared the ex and this It&#x27;s just going to be 12 pie get from integrating the constant term 12 pie minus nine over four times 39 over 12 times three. Cute so times pi.

Problem

Find the indicated volumes by integration. Find t…

Problem 28 Easy Difficulty

Find the indicated volumes by integration.
Describe a region that is revolved about the $y$ -axis to generate a volume found by the integral in Exercise 27.

Answer

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Grace H.

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Allyn J. Washington, Richard S. Evans

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