for-the-following-exercises-use-the-theorem-of-pappus-to-determine-the-volume-of-the-shape-rotating-

Question

For the following exercises, use the theorem of Pappus to determine the volume of the shape.
Rotating $y=m x$ around the $x$ -axis between $x=0$ and $x=1$

Amy Jiang · Accepted Answer

Okay, so we&#x27;re asked to rotate. Why is equal to M X across the y axis or the X axis? I mean, So if why is he with MX? We rotate across the white or the x axis. So we have something like this. So it looks like a cone. Okay, So based on this, our area is this This looks like a triangle. So that&#x27;s 1/2. See, this point here is one comma m based on tight. So we have 1/2 1 time, um, is equal to a So that&#x27;s, um, over to our center of ass. If we have a triangle with purchases on these ends, it becomes see, So we have X one, which is oh, plus one plus one. All over. Three men. Zero still plus M J R. Y points over three. This just becomes to over three comma m over three. Okay, Now the distance traveled by the senator. Mess is a circle with radius em over three. Okay, so our distance this becomes to party. Time&#x27;s over three, but you comes to hi. M time over. Great. Okay. And based on Durham of populace, we have a volume of a time feet which is equal to em over to attempt to, um to pie am over three becomes pie am squared over three.

Vikash Ranjan · Answer

listen well. The figure this is drawn on this. This is our X X is this is our So this is our ex existence is over. What? No. Well, drew this as this is why he was MX. And this becomes sorry blown out of this. And this point is our one Come on him on the center of months lies here. This is You want a country on DDE to him upon three. Because we know this points so far. This is Gino Command zero, this is one combine on DDE. This is eso minus one going in. Are that made this one? We have this point. So you really need to find the volume off the gun. First up on Ilia. Uh, cage eagles went about doing one in tow, so this is equal. Do came upon it. We can take this point. This is also a point which have been courting at that. Do you don&#x27;t? Mm. So really off this triangle, ese. And I do no as a sink it off. Muss is, uh, Therefore, we can see the fact that is center of masses. A friendless extra bliss X tree, a bomb three end. Why wouldn&#x27;t lous y two plus y three upon three. So this becomes, do you know, plus zero plus one upon today. On it is judo plus m last him 13 So we are in the center of months as one upon tree on DDE to one tree. No, The in distance day is Germond by the center of Mars which is the so called it Good radius one of 13 Does these equal to pi into one upon three people Do 2593 their food. Bye The to them off. Okay, Taboos we have They warned him off the going as we waas Really there is equal Do am I want doing do to fire 13 This is equal do in my country upset the falling on it is answer given problem.

Mauricio Araiza Canizales · Answer

in the following problem, we went to the German. The volume off of Soli generated, but would rotate the line bicycle to M times X around the access bicycle to zero or the X axis for ex emitting 01 by using the theory of purpose. So this theory says that the volume is equal to the area of the region that has been rotated comes the distance that the center of mess is translated. That so in this case, little stroh but for a sketch for this region. So we know there&#x27;s gonna be a just a straight line slope. And so this goes from the 0.0 comma Sarah to the point one coma m no. So this is a region that we want to rotate. And so this serum says that we need to find first theory of this region, so let&#x27;s find it. So the area of this region, it&#x27;s gonna be just the integral from IGA V of FX, the X In this case, this area just basically went to me the integral from here to one off m x, the X, which is just gonna be m X squared debate about you that waited from Syria to one, and this is just em. Divided picture. We have found one of the first variables. Now we need to find first the coordinates for our center of Mass to determine then the distance. That is gonna Trent, that it&#x27;s gonna move through just fine to find these in your region. So, first of all, we know that this region it&#x27;s gonna be a cone says we&#x27;re rotating around the line. Weiss equal to zero. So is this shape of the region that it&#x27;s gonna look after. We have rotated it. So now let&#x27;s first find the coordinates for the X and Y X for the coordinates of the center of Mass. So we know that the X and Y coordinates are located. That X is equal to the moment, would respect it. Y Axis debated plate M. And why? Which is the moment with respect to the X axis? Be better. But I am. So let&#x27;s first find M, which is the mass of the system. This is the integral from A to B times the density of the of the system terms FX minus dfx the ex in this case JFX its simplest zero. We can rewrite. This is the integral from 0 to 1 of Rome temps M x, the X which if we integrate, we&#x27;ll find out that this is just em Temps role. You&#x27;re gonna bite you now. Now let&#x27;s find the moment. Would respect to the Y axis This is the integral from A to B of road Tim Sex Tim&#x27;s ffx my mischief x jfx is again the lower function and in this case, is just a line White schools zero So we can write This is the integral from Sergio one of row temps X times Emma Em effects Sorry, M times X, The ex which is just gonna be dating two girls from search one of road James M Times X squared DX So the moment would respect to the y axis Just gonna be a row temps en times x cubed, divided by three Which is the interval off this term break here Evaluated from Syria to one which it turns out to be a row terms m divided by three. Finally, let&#x27;s find out the moment with respect to the X axis, which is 1/2 the integral from eight of&#x27;em off road Temps F Square affects minus G s wear off X the ex we can rewrite. This is the integral from search one have the integral from Sergio one of room temps. Mm X square the X If we actually integrate this jam right here will end up with Ro temps MX and squared, divided by six. So now we can approximate the coordinates Where the ex for the a sense of mystery. So X is gonna be robe temps and be ready by three. You added by M Temps room divided by two. So this is just rode temps and attempts to be ready by row temps am gravely three, but she&#x27;s just 2/3. So the center of mass has exc ordinate of 2/3 our white cord. In it is Ro James M iss where but it by six Divided by and Tim&#x27;s Road derided like to this is just row temps. Emma squared camps too divided by six times in Tom&#x27;s room. Well, she&#x27;s just am divided by three. No, let&#x27;s take a look back. So we have our center of mass located 2/3 do co meth am divided by three. So let&#x27;s say that this is the point of the center of mass knees in our color. Actually, when you scream So let this point B since you&#x27;re a mess, once we rotate this interview mess, this is gonna travel. When I run a circle, I run a circle with radius equal to M divided by three. So to find the distance, we just need to find circumference of the circle. So the distance that this interview mass travels, it&#x27;s gonna be equal to two pi times and the right of way through, which is the radius, which is which now trends into two pi temps chimps m divided by three And finally using that your purpose, we can actually find the volume off this solid generated by rotating the function y sickle to annex around. They were line bicycles, bicycle to zero. So we said that this secret to the area of the region comes the distance that this interview must troubles. So this is gonna be m divided by two times So this girl will be multiplying. It attempts to pi times m divided by three, which is just pie. I am a square. You read it by three. And this is the solution to this question

Amy Jiang · Answer

Okay, so we&#x27;re asked the boy in the volume of the ship created when rotating on the curb from 1 to 2 around the X axis and was given the picture. It looks like this, and we have wise equal to Ellen effects. Okay, so our volume is equal to high time to interrupt from 1 to 2 of our function, which is Ellen of X squared. So what do we have but the owner of that, That&#x27;s the people to ex ln to exploit its too integral of Ellen of X, the X based on innovation by parts if we let you equal to Ellen Squared and e u equals to L. A X, we have the vehicle T i s and B X. All right, so we get this and then we need to integrate this portion one more time in a go of this. Is this Ellen of X times X minus X. So we get X owen squared of X minus two expo and X minus two x and it needs two times this by high and an integrated from 2 to 1. What does this give us? This is just buying times to Ellen squared of two minus two before Elena to when it sport minus high times alone. One. When it&#x27;s two alone of one minus two, he&#x27;s become sterile. We get negative to pie. Okay, so this becomes just be combining this. We get to pine. Actually, we get this five times to Ellen to wear, to punish or relative to us, too. Wait. Minus. Only should be plus. Sorry about that. Yeah, get this. And then, in fact, attitudes with two pie. I&#x27;m alone. It&#x27;s Owen squared of two minutes to a lot of two plus one.

Adrian Fan · Answer

in this problem were asked to find the volume of a solid of revolution just form by rotating the curve occur of y equals natural log rhythm of X around the x axis and the bounds will be X equals one or two. There&#x27;s going to be rotating this region around the X axis. So we&#x27;re going to use the washer method and, you know, I disconnected and discreetly you can see it at the area and the volume of the resulting solid should be high friends integral from wonder truth of natural lager than of X squared DX. So this is the integral expression. However, we actually have to evaluate this integral. So for those of you, uh, so do you drink it? And it requires a medical integration of parts buy parts. So if you don&#x27;t know that method, uh uh, where you can do for now, just put this in a row on your calculator and get a decimal approximation. But for those of you do know integration by parts, I can show you how we can actually do this integral. I wanna begin with an integral gonna go from one or two pipe has able from wonder to of natural longer than X squared DX So going to let you Nicholas National longer than of X on DVD equal d x or so you equals natural order that X squared than devious DX So d&#x27;you is going to be too natural longer than X over X dx and then he is just going to be equal to X. So that means this Integral wants to use the congressional part is going to be you ve that&#x27;s X had natural longer than X squared minus in agro of VD you That&#x27;s the integral of two natural log rhythm of X over x times X so the exes can&#x27;t slap for just left with integral to natural longer than X t X. Yeah, I&#x27;m running out of space, but it&#x27;s going to be this expression. It&#x27;s what are integral is simplified to I should be an equal sign. So next week, to evaluate the integral natural longer than X and that will require integration by parts again. So this is our expression. Now all we have to do is calculate integral natural log rhythm of X t X, and then we would be done. So this is a simpler, integral. So again, these integration I parse thatyou equals national algorithm of X and DV equal d x, then do you? It&#x27;s going to be d d X over X, but then he is going to be equal to X. So this integral is going to be all the UV. That&#x27;s exponential lager than of X minus in a group of Edie you well, the X over x times it&#x27;s just d X. So the X&#x27;s cancel out Luckily, and the integral of D X is just, uh, now just vehicle to X. Yeah, so this integral is going to be ex natural lager than X minus X plus c bear the cost of integration. So if we substitute that back into here, now&#x27;s our expression for Oh, it hurts. This should be an indefinite Integral. I&#x27;m sorry. So nope, I hear no bounds during the integration with parts were goingto plug in the bounds. Later on, we&#x27;ll actually find a record Valley that in a movie right now I just care about the indefinite integral the anti derivative and then we can calculate the definitely Nicole once we know the anti Drew. So I&#x27;m sorry. This should be an indefinite integral oversight on my part. But anyway, we can finally indefinite integral of this now because we know the anti rid of natural longer than of X. So is summery, uh, at another page. You know, the indefinite integral of natural longer than X at square DX will be equal to we have Here is X times natural longer than X squared minus true ties that indefinite integral natural longer than of X t x. We know the indefinite integral of that is just x his natural longer than X minus X We need a plus C at the end. So this will this&#x27;ll be re reaching after factoring and X as natural of x times a quality natural order The X squared minus two national level of X plus True, they were plus C Oh, that&#x27;s the indefinite, Integral. Well, we want to calculate was the definite integral from one or two and then multiply that by pi. So that&#x27;s not too bad. So the final answer is gonna be pie hands that expression evaluated at two. So when x equal to tube, the expression is going to be able to have the quantity actual log of true squared, but it&#x27;s too natural. Article two plus true my and then with this attractive value of the expression when X equals one. So that&#x27;s going to be one of the quantity. Actual log of one squared after lager 10 So that just zero when his true natural log of one natural look of 1070 if I have a truth. So after some simple occassion, because rewrite this as Troop I under quantity after a lot of true squared minus two natural log rhythm of two, then plus one, and that is our final answer.

Mauricio Araiza Canizales · Answer

in the following question, we went to the chairman The volume of the solid generated Once we rotate the region in between the functions Xs equal to two y squared minus y que xs equal to zero. Why&#x27;s it called zero? And all of this is being done around the X axis and the procedure to calculated volume must be cylindrical shells. So first of all, let&#x27;s determine 0.0 comma y for all of dysfunctions. Sorry for this function over here. So we blood in zero. We&#x27;ll have to wake you. Minus Wait. Sorry to waste square miners. Wake you now would spool a Y square. That&#x27;s a common fracture. Will have to minus y out of these will have to solutions Bicycles zero and to Y minus physical zero to minus Y is equal to zero. Sorry, this has a solution. Why? Sickle to two. So you have two points. Zero comma, zero end zero. Commit you. So now let&#x27;s determine the point X comma zero. In this case, we&#x27;re gonna sit. X is equal to zero. Sorry. Everything on the on this site that&#x27;s gonna be zero. So in this case, we&#x27;ll have the 0.0 comma zero. So our bounds in this case are given by only these two points. Zero comma zero and zero commit you. We&#x27;re gonna perform cylindrical shells utilizing Why? So our bounce in this case is gonna be zero m b is gonna be too, you know, it&#x27;s actually right. The integral the volume in this case, your license Electrical shells is gained by two pi chimes integral from eight of you off to pie temp. Sorry off. Why Times x of y b y, in our case, are integral is gonna be to pie Sorry toe by Thames an integral from 0 to 2 of why times to y squared minus white kids d y These can also be written us to pi times the integral from Syria to two of two white cube minus white to the fourth d Y. And this is simple. Integral to ballots will have to pie times the integral of to wake You won&#x27;t have y to the fourth minus the integral of why Fourth is way to the fifth power divided by five. All of this is the ballet that from sure to two we have two pi times zero evaluated over here is just eight minus 32 divided by five, which is just 16 pi divided by five. And this is the volume on this region and does a solution to this question.

Problem

For the following exercises, use the theorem of P…

Problem 280 Hard Difficulty

For the following exercises, use the theorem of Pappus to determine the volume of the shape.
Rotating $y=m x$ around the $x$ -axis between $x=0$ and $x=1$

Answer

$\frac{m^{2} \pi}{3}$

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