find-the-volume-generated-by-revolving-the-regions-bounded-by-the-given-curves-about-the-x-axis-u-23

Question

Find the volume generated by revolving the regions bounded by the given curves about the $x$ -axis. Use the indicated method in each case.
$$y=3 \sqrt{x}, y=0, x=4 \quad(	ext { disks })$$

Gregory Higby · Accepted Answer

thing with this problem is that we have the square root function but it is tripled. So it is a A vertical stretch by a factor of three. But then we also have the lines, Y equals zero, Which is why is zero then x must equal zero. Uh You know, three times zero is zero. I hope that makes sense because they also give you the bound Of X. equals four. And so as I do the disk method, as they tell me to do the disk method, uh it&#x27;s pi times the function squared. Uh in terms of X because we&#x27;re revolving around the X axis. So that&#x27;s why my bounds are zero and four is a as I mentioned, my ex zero but also they gave us the bound of four and then it&#x27;s a function that they gave us. Now if it&#x27;s helpful out encourage students, especially if you&#x27;re afraid of making a mistake of actually squaring it. So three times serious nine and road X times X is just X key X. So at this point you can evaluate the integral by adding one to the exponent and divide by your new exponent And do that from 0 to 4. And uh I don&#x27;t know, I think students should be able to do four squared in their head of 16 divided by two is eight times nine is 72. When you subtract 00 squared? Still zero times nine half. Still 0. 72 minus zero is 72 times pi 72. Pi

Adnan Gill · Answer

we have to occur. Why is equal to squared off X, which means X is equal to y squared And then we have our line X is equal to four. So this point here would have coordinates off for two. And the area that we wanna go take a body. Why accesses this area? So and this we take this element of this for his start your case are or the inner radiance that would be equal to Why Square and the upper case our people to four so frumpy. Why access now? The volume off the solid generated would be in a girl from 0 to 2 pi times r squared minus R squared d y or he&#x27;s equal to high kind of the integral from 0 to 2. Big case are is force of four squared minus y squared, squared. Do you know why so easy? Going to high times the integral from 0 to 2 16 minus y power for Eli Musical to five times 16. Why minus wide to keep our 5/5 mitts off integration from 0 to 2. So we is equal to high times 16 times two minus two to the power 5/5 minus zero minus zero. So we have a B is equal to pi times 30 to minus 32/5 or me a physical 2 128 pi over five.

Adnan Gill · Answer

So why is he going to seek and off X Which means why square we&#x27;re people too. Secret square defects. The volume would be cool to any grilled tie. Why square the X from zero to fight over four or volume is equal to We&#x27;re gonna put the pie in front so five times integral from zero to high over four seeking squared x dx The integral of Seacon Square X is changing decks so we have V is equal to high and change index with the limits off integration from zero to high over four so t would be cool to pi times changing off pie or four minus changing off zero and Kenya compile for is one changing of zero is zero so the water is equal to pi.

Bobby Barnes · Answer

we want to find the volume of the solar generated. When we revolve. The region X is equal to over a square of wide plus one exit with zero y z zero and wives three around the Why access. So verses go ahead and write down what formula we want to use for this. So since we&#x27;re rotating about by access, we would want to use some interval from A to B, which will be our pounds of integration, some radius function r squared. And in this case, they&#x27;re gonna want in terms of why we&#x27;ll need to multiply this by high and will integrate with respect to why let&#x27;s go ahead and just kind of color in the region that they said we&#x27;re interested in. So it said from why is equal to zero by three. Well, that would be here here as well as X is equal to zero, which is here. We&#x27;re going to take this region and take some point on it and rotated around. So it&#x27;s going to be this here, and we rotate around B y access like that so that green line that have grown there should be our radius. And in this case, That would just be the distance. We are along the X axis and we&#x27;re told that distance is just too over why plus one and all of that square with So we have our function are in terms of why and now we can just go ahead and plug that in for first up. One of the things I&#x27;m gonna do remember, we can pull that hi out in front of the integration and then we&#x27;ll want to square two over this. Where the y plus one. So in the numerator we&#x27;d end up with or an end of the non there. We just have wide plus one when we still have our d. Why Over here now we need to figure out what our balance of integration are going to be. And lucky for us, we don&#x27;t really have to do much because they already tell us that. It just needs to be from why isn&#x27;t with 02 three So about zero here and three there. Now, what I&#x27;m going to do for this to integrate it is using U substitution, so weaken right on the side. But you equal live plus one, because this kind of looks similar to the integral for natural log. So if we make this substitution here, we can go ahead and take the derivative and get the differential. D&#x27;you is equal to Dubai and then we can just start plugging things in. So first time to pull before out front. So I get for pie here. I can go ahead and replace why, plus one with you. So one over you and d y just becomes now. The last thing we need to do is figure out what our balance and expression are going to be. So we have when why is equal to zero? Well, plug it zero into that will give us you isn&#x27;t one. And when? Why is it it&#x27;s three for getting into that equation? Can we get you? Is four so we can go ahead and put one or here and now we can integrate. So we get four pi won over you into role bins, Natural love of you and then we want to evaluate this from one so we&#x27;ll have four high natural log of four. My next natural log of one and the natural log of one is zero. So we&#x27;re just left with or pie natural log of four for the volume. When we rotate around the Y axis

Russell Arnold · Answer

Okay, so here said typical um disc method alarm volume solutions. So the bounds of integration are given negative pi over four two pi over four and the you&#x27;re a pi r squared r is second squared x So come the anti guerrilla who seek out squared acts is tension tax. So we get a pi times tension by over four minus attention of negative pi over four. So that is just going to be, um, to pry, because each of the is this one of those negative ones survive.

Adnan Gill · Answer

So our curve here is X is quick to discredit off four minus y squared. And in the elemental guest here, the radius R is equal to s And the thickness off course isn&#x27;t going to d y. So we and the limits off integration are from negative to to positive. So we cannot say volume is equal to any girl from negative 2 to 2 pi r squared D y or vehicle to any girl from negative to to to. And we can put the pie in the front so R squared would be four minus y squared under the square root squared deep That would give you why is he called to high times integral from negative 2 to 4 minus y squared g y or these equal to five times four. Why, minus y to the power 3/3. The limits of integration would be from negative to do positive two. So V is equal to the high four times two minus two to the power three for three minus four times negative. Two minus negative due to the power. 3/3. So musical too high times eight minus over the minus negative eight plus eight or three. So simply find that the is equal to PI. Times eight minus 8/3 plus eight, minus 8/3. So simplifying the parentheses gives value is equal to 32 high. What? We&#x27;re 30 here.

Problem

Find the volume generated by revolving the region…

Problem 9 Easy Difficulty

Find the volume generated by revolving the regions bounded by the given curves about the $x$ -axis. Use the indicated method in each case.
$$y=3 \sqrt{x}, y=0, x=4 \quad(\text { disks })$$

Answer

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