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

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

Related Courses

Calculus 1 / AB

Calculus 2 / BC

Calculus

Chapter 2

Applications of Integration

Section 6

Moments and Centers of Mass

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Video Transcript

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's determine the shape of this graph. So we know that we'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're gonna substitute are by a we'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'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're gonna use. We're now on. It'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'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'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's gonna be equal to the area of the region that it'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'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's first calculate the mess. So Emma Sickle to the integral from a TV of Rome times FX minus your effects the ex We're here. We'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'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're right. It'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're going to determine the moment with respect to the Y axis, which is just an integral from a TV of Roe Jim'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're here. I wanna substitute Charlie Mintz. It'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's just Juve room temps a square of minus X where we'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's Road Temps pie which in this case, this terms cancel this one and this one we'll have for a you write it by three pie. So no, let'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'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'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'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's determine the volume. We'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'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

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