for-the-following-exercises-use-the-theorem-of-pappus-to-determine-the-volume-of-the-shape-a-gener-2

Question

For the following exercises, use the theorem of Pappus to determine the volume of the shape.
A general cylinder created by rotating a rectangle with vertices $(0,0), \quad(a, 0),(0, b),$ and $(a, b)$ around the $y$ -axis. Does your answer agree with the volume of a cylinder?

Vikash Ranjan · Accepted Answer

Jim, Just a phone. If the coordinate the Joan this is our ranks is it is our axis and goes You don&#x27;t No, We have this as Zito. This is as Come on Joe does is Come on, baby. This is Juridical Monty. So we have the center of months. Nice What he owned kisses Same drug mask for cheese. If I do on DDE, we have to be right. So first of all, we have to proof that how doesn&#x27;t drop mask? It&#x27;s like this. So we have the formal loves and drove miles as you&#x27;re a blast A plus it lasts Gero open full on. And this has Gina plus direct last p plus be apart for now this is equal. Do you were born for Andi? We have to be upon food says equal. Do opponent land be a part of. We have this in drug mass at this? No, The distance Jones by the center of Mars is the circle Wait radius! You do dust d equals Dubai here. Well, two This is evil Bye. No, it won&#x27;t take him off the booze Ward him off. Sitting down is equal. Do really is equal to you. He being do here by this is Eagle s Squad B. Bye to Cece Dancer to a given problem

Amy Jiang · Answer

Okay, so we have a triangular shape around the Y axis. So let&#x27;s say we have something like this and we&#x27;re asked to rotate it. Yeah, I saw Area is equal to 1/2 our length over here. It&#x27;s got us a and our hides over here. Let&#x27;s call to speak. Okay, so that&#x27;s 1/2 a times B. It becomes this a b over to tell if a triangle hazardous ease from this point here just once here in this place yet actually, we only looked from Joe to these Deep wants. Our center of mass is the sum of our X points, which is Darryl plus a Let&#x27;s start all over three and some of our white points, which is open. So being over three, right, because we have our verte sees at socom a job from a comma Joe and become a job. So this becomes a over three comma, be over three, and now we need to find our distance. But our distance is this equal to two high times our export, which is a over three. So that&#x27;s just to a pie over three. And based on the dam of purpose, a Times T is equal to our volume, which is equal to a B over three over two times to a pie over three, which becomes a squared B by over three.

Mauricio Araiza Canizales · Answer

in the following problem, we went to a German. The volume of the shape generated by rotating the triangle with practices at Sirica Macero, a commissary and serial comma be out from the y axis. And we want to find the volume by its in the theory of steps. So in this case, what goes in the first start off by finding the sketch of this shape So we know it has a purchase. Critics located zero comma zero, another one in Syria committee None. There were not a comma Syria. So it&#x27;s gonna be the 0.8. Come on, zero. It&#x27;s when the 0.0 come up. These are not drunk to scale, by the way. So this is a triangle that we want to rotate the bronchi y axis. And if we do so we&#x27;re gonna find out that we&#x27;re gonna make a cone. So let me just read, draw these. So we&#x27;re creating a cone by rotating dis regional here. And so we want to find the volume using the serum of Pappas which states that the volume is equal to the area of the region. That it&#x27;s been rotated comes a distance that this interview mass travels around this region, So let&#x27;s first find out the center of Mess. So this interview mass for our wreck tranquil right here with three purchases, has X coordinate of except one plus except two plus except three. You write it by three coma wakes up one plus waist up to was wasted. Three divided by three. If we substitute the valleys over here, we&#x27;ll have a very big three coma. Be divided by three. That&#x27;s a recording this for our center of mess. So in this case, this point is going to be located roughly a run here. It will rotate this point around the same axis. We&#x27;ll find out that he travels around a circle, so the total distance that it travels it&#x27;s going to be equal did the circumference of the circle. So we know the desert conference can be obtained by multiplying two pi times the radius. In this case, the radius is gonna be the distance from the X axis, and it&#x27;s located at a coma 38 divided by three units from from the origin. So the distance that it traveled this distance of the center of my struggles. It&#x27;s gonna be to pie times a divided by three. It&#x27;s just just simply do pie a bite of a three. Now let&#x27;s find out their area of the region. This region is a rectangle, so we can actually use the Formula 1/2 the base attempts to height which in this case will have 1/2 times eight, which is the length of its base temps B, which is the length of its height. Just just eight times. Really. But I&#x27;d like to. So finally, our volume is given by the area of the region, a temps we divided by two attempts to pi terms. A delighted by three. This ends up being yes, a square times be comes by divided by three. So this is the answer to this question. And before we&#x27;re done, letters just confirm that this is our answered when other the volume for any cone is just 1/3 of pie. Attempts are squared times height. So over here, the radius is gonna be eight in our high. It&#x27;s gonna be be we&#x27;ll have one hat, 1/3 off pie times of radius, which is a square times the height, which is just be. And in this case, we have found out that a squirt him speak time spying divided by three. The same result there. We have obtained using the theory of tapas so we can confirm that this is the correct answer by using geometry.

Vikash Ranjan · Answer

the problem. Just perform. Destroy this graph just like this. Okay, it is a circle. Now this circle there is a drug mask here. This is a center of mass. It is fully upon three by on it is you. Actually, it is not a circular shape. It is simpler. It is like that, Bruce. But it is actually a spectacle. Yeah, three day long. So this is our Why, which is Kodoroff? It&#x27;s for my next exit squad, which you don&#x27;t on Dhe. We have seen this is us interrupt Mass. Uh, this is our Why which is equal to minus the door off its one minus six Esquire. This is G X. No, we will see that. Yeah. Uh, good. This fair is equal to a which is equal to how, by Esquire. This is Eagle Do is by a bomb, too. No center mass, that is. He&#x27;s given us. It&#x27;s far from a jet. Oh, so this is expanded. His equal do in village. It&#x27;ll do a X off fixed my eyes. Ears, Dix. Fois ce. Yeah. So this is eagle do indeed. A general way. Thanks. And a wreck it Whatever off is minus six squad minus and minus of Vukovar is bad manners. Thinks it&#x27;s quiet. D x o r. Yeah. Now this is equal Do and you don&#x27;t You don&#x27;t do a two exit door off isco R minus 60 squad DX over it. We gonna do minus minus. Integral D o a minus six under wreckage is by the minor success while to the Barbara off on a pawn to deals over it is the will do minors off the records to buy three into Esquire minus x Esquire The power of three upon to Joe Do a over A No, this is equal. Do minus record. General minus 213 You or a legal do we went to be Thank you. For what it No, this becomes to my three. Thank you. A phone iss What? Buy you one toe is illegal. Do toe you my tree in tow To mine. Yes, but final visit even do for upon three ever. Hi. So we have three upon tree pine? No, no distance day traveled by the center of mass is the circle wheat. Really? Yes. For your 13 bank. Therefore be waas to buy Indu for you on three by this is equal. Do in here by three hands to them all. But who is we have wanting more. It&#x27;s fear, Which is it, baby? On this equal quests warning which is equal do case for Why over do in tow India one tree This is even tow 413 you buy. This is an answer to the given problem.

Mauricio Araiza Canizales · Answer

in the following problem, we went to the German. The volume of the shape generated after we have rotated a semi circle of radius, eh? Around the y axis. Introduce you the 12 years If he erm perhaps so. First of all, let&#x27;s determine the shape of this graph. So we know that we&#x27;re looking at a semi circle and any circle has your question. X square was White Square is equal to our sward. In this case, we&#x27;re gonna substitute are by a we&#x27;ll have the equation X squared plus y squared is equal to a square. We want to rotate this semicircle around the y axis. So we&#x27;re looking something on either thes two sides. And since we know that this is a circle, we can draw the craft like this. So this is the semicircle, since since we have the German, that is gonna be the graft that we&#x27;re gonna use. We&#x27;re now on. It&#x27;s just sign unquestioned for descript, which is gonna obtain which is gonna be Y squared is equal to a square minus X. Where and then we&#x27;re gonna take the square with the both sides. Have you square with the face where my sex square notice right here. That the positive sign of the square. It gives this half of the graph, so we need the positive and the negative sign. So this is the region. We&#x27;re gonna rotate around the Y axis joke, Tina. Complete sphere. So no. The theory of poppers said that tested the volume of this region. It&#x27;s gonna be equal to the area of the region that it&#x27;s been rotated times the distance that the center of mass travels during that rotation. So we can actually find the area of the region by identifying this as a semi circle. So the area faster complete circle is just my are square and that for the semicircle, is just being development. You. So we&#x27;re here. The area of the region is gonna be pie. Hey, you swear you better like you now when it do determine the center of mess. So we know that the coordinates of the center of mass are equal to the moment would respect to the Y axis you fight it, fight em. The moment would respect to the X axis dependent. So we first need to identify over here that the region has symmetry with the X axis. So over here this region right here has symmetry around this. So we can say that this section right here is the same us, this one. And because of the symmetry, we know that the white coordinate for our sensor of mess must be at the y sickle zero. So the only thing that we need to calculate is a moment with respect to the white access, the total mass of the system. So let&#x27;s first calculate the mess. So Emma Sickle to the integral from a TV of Rome times FX minus your effects the ex We&#x27;re here. We&#x27;re gonna integrate this from zero to a us row Times square with the face where minus x elsewhere plus square root of a square minus X s word were substructure in negative sign of discovery which just give us a positive sign. So this just becomes twice the integral from Syria to a of Rome. Attempts a square root off a square minus excess where the ex if you use a graphing calculator computer software to find this integral, you&#x27;ll find out that this is a multitude of temps room terms, a square terms sign members off X. You write it by the absolute value. A You&#x27;re right. It&#x27;s true. Plus extends to square with the face Where a square minus x s word absurd for that excess. Where do you write about you about? Waited from Syria. This gives you a massive row times a square camps pilot ready that true divided by two which is simply just a squirt and throw attempts pine He raped you. So now in another wide world, we&#x27;re going to determine the moment with respect to the Y axis, which is just an integral from a TV of Roe Jim&#x27;s ex Times Square with the face square minus six square plus square with the face where in minus x X square Yikes! Thes again just becomes the integral off two times the interval from Syria to a Oh, sorry we&#x27;re here. I wanna substitute Charlie Mintz. It&#x27;s well, have syrup to a sure to a temps row terms ex times the square root of a squared minus X square. The ex, by using your substitution, will find out that this integral it&#x27;s just Juve room temps a square of minus X where we&#x27;re through three camps derided by three Val waited from Syria to rate the value of them off. The moment with respect to the Y axis is just to a cube temps row divided by three. You know, we can determine the x coordinate. Just going to be tu es que tem strom you better by three divided by in square times roadie but time spine divided by truth. This simplifies to to a cube times road temp stew you branded by three a square Tim&#x27;s Road Temps pie which in this case, this terms cancel this one and this one we&#x27;ll have for a you write it by three pie. So no, let&#x27;s go back to our equation right here. It or it growing. So we have the chairman that Gordon Surfer, Center of Mass. So let&#x27;s suppose at the center of France is located at this point right here. When we wrote it around the y axis, this interest masters coin goes through a pat that resembles a circle. So in this case, the distance that the center of mass is traveling. So this distance of this interest mess, it&#x27;s going to be Dad off the circumference. Sadistic conference is going to Pike Temps are this case far is a distance from the origin, which we have the chairman to be. The X coordinate off the center of Mess, which is for a divided by three three patisserie. So we&#x27;ll have troop ie temps. Art Sorry is equal to two pi times for a divided by three pine, which is equal to eight a divided by three. We have already determined the area of the region. So now let&#x27;s determine the volume. We&#x27;ll have pie tempts a square the right of the truth times four times a day, degraded by three, which is simply just 4/3 times pi times a cube before finishing. Let&#x27;s just test the pieces too. Correct answer. So we know from geometry that the volume of us any sphere is just 4/3 times pi attempts are acute and we know that the radius is our A. So we have 4/3 high times a cube, which is the same answer. So by using the theory of apples, we have the chairman the volume of this year. We have confirmed that is the correct answer by using

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.

Problem

For the following exercises, use the theorem of P…

Problem 283 Hard Difficulty

For the following exercises, use the theorem of Pappus to determine the volume of the shape.
A general cylinder created by rotating a rectangle with vertices $(0,0), \quad(a, 0),(0, b),$ and $(a, b)$ around the $y$ -axis. Does your answer agree with the volume of a cylinder?

Answer

$a^{2} b \pi$

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$a^{2} b \pi$

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