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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.
$$y=\sqrt{x}, x=0, y=2 \quad 	ext { (shells) }$$

Gregory Higby · Accepted Answer

it makes perfect sense when you look at why? Because the square root of X. And that they want you to take the region between X Equal zero. Um and y equals two. So we&#x27;re talking about this region in here, it makes perfect sense to use the shell method as we&#x27;re revolving it around the X axis. Um so what&#x27;s the show method? It&#x27;s two pi the radius times the height. Uh In terms of why, why is it why? Because the radius of each one of these cylinders will be why? Units? Um Now when X equals zero, the square to zero gives you zero for y. And they also give you the other bound of Y equals two. Um Now the other piece that sort of tricky as you want to solve for X, which is actually just why squared? Um As we do this problem. So I think you could do two steps at once. For instance why times Y squared is why cubed. And then when you do the integral and you add one to the exponent and divide by your new exponents. Um that we just did two steps at once. And what I also think it&#x27;s very easy about this problem is two to the fourth power or two times two is four will cancel out with this for So you&#x27;re just left with two more to the left, which is just four. Are you do two to the fourth power? 16 divided by four is four. And then also the bound of zero, this gives you zero. So four minus zero is four times the two pi gives you a pie, which is the correct answer.

Abigail Martyr · Answer

we&#x27;re putting the volume of the other cream. Should be Rick rotating the shifted regional Boulder excesses. We are using a cylindrical show method. And since we&#x27;re integrating with respect to why, then we need to make sure that I will. Equations are in terms of way, So oh, in question. Why? Quote two minutes x Don&#x27;t slow this thing. My saying X is equal to two minutes. Why in the equation one equal skirt a bit? Someone saying X is equal to y squared? No. Now that we have this done, we can go ahead. And I didn&#x27;t defied the radius from the height The radius We&#x27;re being simply equal to why, from the height goes from left to right. And that would be the difference between the question on the right. With the collusion on the left that would be minus wine Mine. Why square? It&#x27;s a good volume be given by the equation two pi times that integral from white equals zero two y equals one of the radius Time to the height And that works out to be too. Why my notes Why squared when we why cute Do you want integrating? Yeah, white squared money is what? You know her three. It&#x27;s what you do. Fourth of four devaluing from like was there toe where it goes. One. You have to find time, Warren. Mine 1/3. My mother quarter at y equals zero. This gives them Seaver, so we don&#x27;t Well, Reebok, anything else after this that we are only just attract what this gives is five well and sort of volume and being bye bye over six.

Abigail Martyr · Answer

we have finding the volume of the solid number Teaching good She didn&#x27;t region a bone to x record. We are using this ventricle shown method on We are integrating with respect to y. So we need to first make sure that along equations are in terms of wine. So here we have blood equals like or to discourage a back that is the same as saying X is equal to y squared on the other equation y equals X minus two is the same thing is equal to why, yes to now we need to identify the reading some too high already it is the vertical distance. You you are This is equal to why and they&#x27;re all high as the horizontal distance from this story So we can find the height right during the equation under right, Which is why close to tracking the equation on the nectar. Why squared volume being would be equal to two pi times The into girl from y equals you know, like one too of the radius times the height. When you multiply that out when we get negative Why cute plus two y squared after a plus wife Great. Not too high school. It seems to me y squared plus two y do I When we integrate, we have there. Negative y to the fourth over four. Just watch of the third over three. All right. Why squared? Being evaluated from 0 to 2? Well, you have to kind times key putting two here. We have never 16. A book more. This is eight over. Three more. So what we have is negative. Poor Listen. 8/3 less for eggs era. This is all equal to zero, so you can stop here. So we end up with two pi or a negative full cancels out. Eight of with three. Gives the bond among 16. Prize divided by three.

Abigail Martyr · Answer

we&#x27;re finding the volume of the solid flooring. Barring rotating the shaded area boned Invent a graph of y equals skirt Back X equals zero I like was two x minus one about the y axis. Since we are using the cylindrical show method, we are going to first identify our radius on our height sort of radius can see represented by tax on the high you can represent by during square root of it, minus the equation to live minus one and scared of it was just X adorn how power minus two X plus one. So when we find a blogging things that go into two parts terms that integral from exit cools zero two x equal long our times H gives this extra. The three have minus two x squared the X reintegrate we have to prying times to two eggs over Fry X would be to the five house minus. But, you know, excluding Mr the man dishes being to X cubed over three x squared over two, and we re evaluating from 0 to 1. When we build without on, we have to my 2/3 Well, you know, it just gives to pry times Well, we&#x27;re learning the value wouldn&#x27;t had serious. Just zero. So that gives us a volume, uh, every tie over 15.

Abigail Martyr · Answer

we&#x27;re finding the volume of the shaded area. Results about girl effective were used in the cylindrical show method. So for through in its identified the radio and the height yes, we&#x27;re doing Why integration? Our radius would be represented by this and hurricanes here review equal to why and her height would be found by attracting the equation under the left of the graph under left from the graph on the right, that would be good. Why minus negative for religious these goto. Why? That&#x27;s why now we can find born being is legal to to pry times into growth from why equals they want a wife was too key and our radius times are high when he gives us right into three have that&#x27;s why squared. Meanwhile integrating what we end up with this to times y to go Brian path That&#x27;s why huge. Better great three be any value living room euro two. So evaluating at zero if there are so we&#x27;re just going to value it up at two. Right here it is two pi times, all right to skirt of 32/5 that was actually think friend to eight going to or by you ate three come one denominators keeps this 24 squared to buy 40. Our denomination is not as it should be. 15. Not by turned knowing there&#x27;s a common factor of eight. If I take that out, it over 15. We have 16 pie. They burned by 15 times three square root of two. But by in this you the volume just for reference is sick. I will mention that this is a Brooks in the 4.93 times by

Russell Arnold · Answer

Oh, so in this problem, it could help Straw a picture. Sophie graph here, two ton swear it of X. And then why, too, Um, this&#x27;s with y equals two times where acts in this year. That&#x27;s why is to so toothed squared of ax equals two when X equals one. So our bounds of integration are going to be X equals zero X equals one for the volume. So vehicles integral zero one of a backstage Jax and a vex equals pi r squared which equals hi times one pra minus square Root of cracks squared. Um, because high times one minus two square root of ax classics. So hear this equals high times into grow from 01 off one minus two x to the 1/2 plus x the axe and this equals high times X minus for third&#x27;s X of the three house plus X squared over two. Evaluated sure one. So you try times one minus 4/3 plus 1/2

Problem

Find the volume generated by revolving the region…

Problem 8 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.
$$y=\sqrt{x}, x=0, y=2 \quad \text { (shells) }$$

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Allyn J. Washington, Richard S. EvansBasic Technical Mathematics with Calculus

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus