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

Question

For the following exercises, use the theorem of Pappus to determine the volume of the shape.
A sphere created by rotating a semicircle with radius $a$ around the $y$ -axis. Does your answer agree with the volume of a sphere?

Vikash Ranjan · Accepted 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.

Vikash Ranjan · 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

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.

Amy Jiang · 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.

Problem

For the following exercises, use a calculator to …

Problem 284 Hard Difficulty

For the following exercises, use the theorem of Pappus to determine the volume of the shape.
A sphere created by rotating a semicircle with radius $a$ around the $y$ -axis. Does your answer agree with the volume of a sphere?

Answer

$\frac{4 a^{3} \pi}{3}$

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