find-the-volume-generated-by-revolving-the-region-bounded-by-y4-2-x-x0-and-y0-about-the-indicated-10

Question

Find the volume generated by revolving the region bounded by $y=4-2 x, x=0,$ and $y=0$ about the indicated axis, using the indicated element of volume.
$y$ -axis (disks)

Steven Bridges · Accepted Answer

all right. In order to do 71 we&#x27;re gonna defuse in the shell method. So we have the formula over here for the show method based on our limits. They said between zero and four. So we&#x27;re gonna end up having p of X FX. We about to buy, uh, P of X is gonna equal X in this particular problem. And then h of x is our function that we&#x27;re rotating. Uh, so then, after putting those together, we end up with the inner girl from still ever to fight 0 to 4. We&#x27;re gonna go acts in the axe there, so it becomes X squared square root of 16 minus x squared DX. And then we have Formula 38 that we&#x27;re going to use. In this case, A is gonna be equal to four, and then you is equal to X very R s O. Now just filling it all in. So we&#x27;re really gonna have to do tow to start with, and then we&#x27;ll plug in our limits and then multiply by two pie At the end, you will have 1/8 and then again, you is equal toe act. So have acts. Two x squared minus again. A is for so 16 square root of a squared 16 minus X weird plus eight of the fourth. So, in this case, round up with Ward of the fourth and then Arc signed of you Which again? His ex over four. And but then we&#x27;re gonna go ahead and plug in our limits. In order to this, we&#x27;ll plug in both for and zero and subtract. Oh, okay, so we&#x27;ll end up with 1/8 or two times four squared is 32 minus 16. It&#x27;s where of 16 minus 16. So that&#x27;s gonna be zero just great across that out now and then we&#x27;ll end up with plus hostile for the fourth. And then arc sign of one again. Arc sine of one is going to give us zero. Is this whole part part is zero minus 1/8 plug in zero. So we&#x27;re gonna have zero times. So that first part all the way through here gonna be zero s. So then we&#x27;ll be left with Plus, and we&#x27;re gonna end up with the 2 56 Ford in the fourth. Um, park sign of one of zero is gonna give us a pie over, too. We still have the 1/8 in there. And then we&#x27;re gonna end up with one ace of 256 times pi over too. And then we have to still multiplied by two pi. So we have 1/8 times to 56 times pi over too. Times the two pi and that&#x27;s from over here were constant. Okay, so then multiply. Know that we end up with 32 pie squared.

Amit Srivastava · Answer

here Cushion is if c ah, the problems of elation revolving about, uh by axis. Third is y axis and the problem is revolving about bikes. So grab, it will be like this. And if we take on a strip, this is X And this part is the way sub volume generation will be always by excess Square De Wei in Dickerson No, we have given vie Quito X by who have minus and I dont exquisite. So squaring both side by square will be extra square minus four minus x square. So final part full rice square minus y squared X x squared equal to excess Quiet So fellow up excess square will be Colin Rice Square It could tow excl Ever take common one plus y squared Hence excess square will be full by square by one place by school No put this value in the graph It is by excess square d y. So bottom generation will be I will be outside full also will become our side by school by one place Why skirt the way and ah learn a prelim it&#x27;ll be 0 to 1 So finally it is full of I And if we add one and subject one, then question like this. So this is four by and 0 to 1 one plus vice square by one place by square minus, derided by one plus y squared Geritol one. So finally, if you organize this so full by and do I will be by someone minus down in verse. Why? And it is here Cuban. So finally it is hold by one minus 10 was one minutes by very food. So final answer is pull if I and go full minus pi buy food.

Robert Leedy · Answer

Alright. So first of all, we&#x27;re given a poor abdullah or give in our function X squared like was X squared. Rebound by white was zero and X equals two. So we have our X squared function but we&#x27;re bound by X equals two and why equals zero. So we know this distance here is to when we rotate about the X axis that helps us know our radius. But we also know that up here we should be at y equals four. So are vertical radius. Here we take our cross sections is going to be four our maximum degrees All right, but our radius is given to us by the function X squared. So why equals X squared now setting this initial problem of pretty straightforward We&#x27;re bound from 0 to 4 and it is going to be pi times X squared dx pretty straightforward integral mover pie out front from 0 to 4. No, that&#x27;s squared DX. We get pie times X cubed over three from 0 to 4. You type this into a calculator, so we get four cubed divided by three. And that is approximately 64/3. So our solution part A is 64 pi over three. Mrs. Party, part B, we&#x27;re gonna do in white in part B. We are rotating about the line, or why equals four. So we&#x27;re actually rotating about this line right here. And no, if you&#x27;re not actually gonna have a hold so we don&#x27;t have to worry about that inner radius. We just need to worry about this outside radius so you can and go ahead and set that up and we want to worry about it can. But it should equal each other because there is no actual difference. So we&#x27;re gonna dio our function. Why? Minus four? Because that was our line when we get Y minus four. So we have X squared minus four. Far outer radius in foreigner. It is the distance from our ex access to our line, which is simply four units. So now we&#x27;re gonna set this up. I&#x27;m gonna leave the pie out front just to save us a little bit of time. Our radius is going from 0 to 4 of X squared, minus four minus from zero for four. All right, Now, neither of these intervals are too difficult to calculate. I&#x27;m gonna find him really quickly. But your fastest option is honestly just using your calculator. So we&#x27;re gonna get pie Times X cubed over three minus four x evaluated from 04 minus four x evaluated from 0 to 4. They were mostly everything by pie. So I&#x27;m going to type these into a calculator really quickly. Give me one second to complete that. All right. Typing that into a calculator, we get 34 0.13 repeating and we have this almost played by pie, which we&#x27;re going Convert that into a fraction and write the Pyin. It&#x27;s a curve inverting that into a fraction we get 512 pi over 15. Let me

Robert Leedy · Answer

all right. The function were given this time is why I equals the square root of X. And our bounds are going to be Life was two and X equals zero. Why equals two and X equals zero. So drawing this r squared of x function looks something like this. So we&#x27;re bound by X equals zero, which is our y axis. And why equals two which that will mean that we get for out here. Alright, now part is rotating around the y axis. What you&#x27;re gonna do this similarly, Teoh, the X axis accept our radius is gonna are bounds for Integral are gonna be our radius here from 0 to 4 rather than on the y axis. So we&#x27;re gonna go from 04 are function is gonna be pi times the radius squared are radius is given by this function. So squared ax squared DX. Now that&#x27;s actually making her life very simple because you&#x27;re gonna cancel each other out. Always simplify if you can, so we&#x27;re gonna pour pie out 04 and we have the square root of X. So that is just going to give us X dx. All right, so when we already value and we have pie times X squared over two from 0 to 4. Pulling in 44 square to 16. Divide that by two. We get eight and then plug in zero. It&#x27;s all zero. So we end up just getting ate pie for our volume of part A For part B, it&#x27;s a little bit different because we are bound by a different line. We&#x27;re bound by X equals four nerve mounted at pound were rotating around X equals four. So actually rotating now here, which makes life actually not too bad. Um, so we&#x27;re still going to find our outer, which is gonna be four minus r radius. So four minus r radius, which is the function here, and that is gonna be our outer radius. And our inner radius is going to be the distance from our Why access to our line, which is just our, which is our inner radius. Now we&#x27;re going to go ahead and set up our interval problem, which gives us from 0 to 4 are outside pi times four minus the square root of X squared DX minus are integral 04 high times four squared DX type This into a calculator. So again grabbed a graphing calculator on minors. Math nine song and hit math. Nine. Now, I&#x27;m actually going to not type in the pies because that will make our calculator and not give us our answer. I&#x27;m gonna pull that out first. Pull it out and we get I&#x27;m sure before four minus spirit of acts squared DX minus 04 for squared DX. All right, type this into a calculator. So 04 or I&#x27;m gonna four minus the square root of X squared. Make sure you put in parentheses, so we square all of it. DX minus math nine 0 to 4 again or squared. DX ID equals tacular is taking a little bit toe. Figure this out and I get my solution is 34 0.6 repeating pie, which we can convert into a fraction 34.6 repeating man. Fraction me gets 104 pi over three

Amrita Bhasin · Answer

we know that the general formula is gonna be vey is two pi times the interval from zero to of radius times height, deal backs. Now we know using the cylindrical show, Matt, that our radius is exports one and our height is four minus two x. So radius times height were simply plugging in times d of X, which means now we have two pi times the intro from 0 to 2 Now use the foil such distributed method to get negative two x squared plus two acts plus four d of X. Now we integrate. What we&#x27;re gonna be doing is we&#x27;re gonna be increasing the exponents by one using the power method. And then we&#x27;re gonna be dividing by the new exponents as you can see what we&#x27;re doing over here. Now we know our bounds air from zero to It&#x27;s time to now plug end to our integral negative. Two times two cubed, divided by three plus two squared plus four times to minus zero, which gives us 40 pi divided by three

Problem

Find the volume generated by revolving the region…

Problem 4 Easy Difficulty

Find the volume generated by revolving the region bounded by $y=4-2 x, x=0,$ and $y=0$ about the indicated axis, using the indicated element of volume.
$y$ -axis (disks)

Answer

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