solve-the-given-problems-by-integration-find-the-volume-generated-by-revolving-the-region-bounded-12

Question

Solve the given problems by integration.
Find the volume generated by revolving the region bounded by $y=	an ^{3}\left(x^{2}ight), y=0,$ and $x=\frac{\pi}{4}$ about the $y$ -axis.

J Hardin · Accepted Answer

here we&#x27;d like to find the volume obtained by rotating the region bounded by the curves sign and co sign about the line Y equals one. So here&#x27;s a rough sketch of the graph down here and read. This is the graph of sign from Zero power for and then in blue we have the Cho Sang graph. The green line is the axis of rotation. This is the horizontal line Y equals one And then the red and the blue graphs up top those air The reflections After we were rotated this area down here around the line Y equals one So we see after that we do is a revolution around the line y equals one we have ah, this hole in the middle of our solid. So that means that our cross sections which is indicated by this washer right here we&#x27;LL have holes So we have washers. Oh, so we have a formula for the volume in this case also, we obtained the volume by letting the washers move in the ex direction. So our follow ume will be in terms of X. So we know the volume is the integral. See you in a power before times the area the washer, which is pi times the larger radius squared minus the smaller radio square. So we need to find formulas for big R and little r. So it&#x27;s relying the picture for this. So looking at the picture here, we want a large radius which is the distance from the center all the way out to the end point. So this is a big R. So one way to get there is to take this entire distance from the X axis two, the horizontal green line, which is one. And then if we subtract this distance right here from zero to the red graph, then informally, we have equality here. So we take the whole distance, which is one, and we subtract off the distance from X all the way up to the red. And we&#x27;re left over with this big r. So we have big our peoples, the entire line which is one minus this. Why value here? But since this Y values on the red side graph, we could replace why with signings. So that gives us big r and then for a little R the same idea. We want the smaller line right here. This line segments a little radius, but we can get there by taking the entire line segments or the one and then subtracting the distance from zero all the way up into the blue graph. And then we&#x27;re left over with lous little r So we have little r equals one minus y. But this same reasoning is before. But this time the y goes all the weapons for the blue graph. So this time why is co sign? So that&#x27;s how we find big. Our little are in this case now we&#x27;re ready to evaluate the integral. So first we should simplify as much as we can. It&#x27;s quite implode that pie. So inside we have one minus to sign, plus sign square minus one. Then we have a minus minus to co sign and then minus coastlines where so we could simplify this a bit. So let&#x27;s go ahead and we see that we could cancel the ones and we can apply Formulas should be a X there for science. Where and co sign squared. So use those double angle formulas with a pie in a girl&#x27;s ear apart before minus two. Sign eggs, plus then this becomes one minus CO sex co sign of two X over too. We still have this to co sign and then we have a minus and this becomes one plus coastline to X over to. So we&#x27;LL need more room here. So it&#x27;s goingto new new page so we can go ahead and cancel the one half in the minus one half in the previous page. So picking up where we left off, we have negative to sign X minus coastline to X over, too, to co sign X And then we have a new way says we have to times two of these. Let&#x27;s go ahead and just put those together. Erase this. So this is just after simple simplifying the previous expression. Now we could integrate into these so we have to co sign X minus signed two eggs over too, plus two cynics and our end points or zero power before. So it&#x27;s Quentin plug in these end points, so we plug in power for first. So co sign power for is rude to over too sign of pirates who is one? And then we have another two times room to over, too. When we plug in zero Ko san a zero was one. So we have two times one sign of zero zero. So these two terms go away and were we can simplify to get our final answer. Hi. We could combine these radicals to obtain too radical too. And we have minus a half minus two, which is negative. Five over two. So this is our volume, and that&#x27;s your final answer.

Yaw Asomani · Answer

do Try to find a value. Okay. We&#x27;re just gonna use the, uh, by you is gonna be given by by a TV. Okay, Where in the are the intervals and then f squared. Ok, X minus. Uh, Jean squared, OK. And cheese where? X X. This is what we should find. So you have to do is just put in your things. Okay. You know, your G&#x27;s weird is, um, zero. So only f square just left and they putting your intervals. Okay, So what is your intervals? Your intervals is, uh, prior with you two and then three pi over two K supplier. You choose three pi over two, and then your function is, um, four co sign X. Okay. So when you swear, that is gonna be 16 co sine squared X t x. Okay, so you have this one. So, uh, you have to integrate us one particular one right here. Okay. Uh, you know, use integration by parts because it is not a product. It&#x27;s not a product of two functions. That&#x27;s just one function right here. Okay, So how to find a way to convert this one? Okay, Uh, do not forget that consigned to X. Okay. Is um Hussein squared X minus one. Okay, so co sign squared X can be written as okay. It is just trigonometry. Identity UK. You know that thing? A plus b is Cho sang a You know, my assign a sign be That is what we used to get this relation right here, actually, so you can use it to prove for yourself. Okay, so in place of this one, I&#x27;m gonna put co signed to expose one game that I can integrate so simply for me. So this is gonna be, uh, signed Teoh Export for two. Okay, then close X k. And this is no limit. And this is upper limit. OK, so I would have to do is just do this little substitutions. Okay, Just put in upper limit my slow limits. Okay? And you&#x27;re gonna get, uh, So here is to you. Okay. Co sign two x is twice clothes hang squared minus one. So then co sine squared minus one is gonna be this and then over to here, OK? Over two years. So that over two is gonna is gonna take away I za ne divide 16. Kate. So in devices seen here is gonna turn into eight cases when it turns into a right here. Then you can. Then you can now continue. Okay, so our problem it minus lower limit. Okay. When you do that, here is gonna be sign too. Times three pie. But we&#x27;re too right like that. Then close three pi over to you. Okay, then here is also over to then minus lure. Limit limit is also sign, uh, twice times pi over two, then over to here, then plus pi over two canes. So that is your upper limit. Minus lower limits for you. Right here. So once you do that, you can see that this is gonna take away this. And so you have ate pie. They have signed three pie over to Temple Astri off three pie here, Over to you. Okay, then this one is taken away. This one. So you have sign hi. And then all over to okay, right here and then, uh, minus by doing so. Include this one on your calculator. Where you gonna get you get a pie squared? Pay so eight by square. You know, it&#x27;s cute. That is the value because you&#x27;re finding a volume. The volume here is a bisque with you need cubic. Okay. That s t eunice for the volume

Amit Srivastava · Answer

increasing like or two. It is one by one minus sine X does it? We&#x27;ve all being involved The excesses bottom off the solid of 10 by the rebuilding about the X axis. So this is the X axis and problems and recently like this theater case. So we will take strip as by and thickness as DX. It is very from Jeter to buy buy food So volume generated volumes Relation will be will be by why square DX and this is here to buy. So follow it could do dealer to buy food the x so ah ah following generation off this part will be neither toe five Eiffel and the ex if will convert one signed us as to 10 x by one plus 10 square expired to part your half angle formal I will be applicable Ultimately it is volume ical toe by was your sort of buy buy food and this part will be 60 square so we can write it as one plus stance car expert too and Seka Square Expecto DX And based in the denominator, it will be can x by two minus one square park. Now let&#x27;s we&#x27;re going to send the final limit. If you consider that expecto entity the whole valley will convert like this. Let tennis by two minus one as he so now second square tryto one by two the x will be DT. So finally it is. Consider too Bye bye for limited Later on will change bullets. Quick Bressman, Your duty by D squared the thirties by zero to pi by full and he squared plus one plus one into 22 will become here 50 by this question, sort by. And it is finally one plus two y t square duty certain by dirty dirty minus Do bhakti now put the value up ts then expert to minus one minus two by 10 Expert to minus one general by by food. So finally it is to buy then by by eight to minus one minus two back then by eight

Enkhzaya Enkhtaivan · Answer

in the problem. We&#x27;ll also have a volume into rules. So we&#x27;re given the function wise to sign Axe Christ, second of X when excess grand from zero to pie over three. And we know we&#x27;re going toe one of all of this grass. Wrong, um, about the X y co zero, which is the X axis. So let&#x27;s see how this will look like on the graphic gonna have Why x zero hi over three and that zero we have sign of Syria. Plus second, there&#x27;s air, which is one and that high over three. We have sci fi or through per second of pyre with three, which is 1/2 plus, Let&#x27;s see. So why ofthe pie over three as I don&#x27;t have a plus one over co sign of experts to divide by square three. This is something that&#x27;s a little bit bigger than warrants, so it&#x27;s kind of you turn around here. The point is, their function there goes, is always positive and presumably go something like this. And if you want to be re Gorius about why it&#x27;s always positive, you can simply look att the function. So it&#x27;s a sign of X plus one over co sign off Axe. And when Axis Going from zero to pirate to both of this is while signing faxes. No negative except because they&#x27;re zeroes included. And whenever Costa nexus all this positive. So this is all this positive and dear. This method says that volume this pickle too. Then flight times Zira internal observed the pie over three. Why are the way years of squared? But then the way uses simply this this distance that is radius is simply the value of our function. Wax at a given point acts here, so this is simply sign off acts. Our second off X scored the ex and all introduce Compute this interval now and so it&#x27;s expended bracket here. So pie, sir, the pie over three science squared If Rex now we could This is a simple trig integral in two times Sign of X I&#x27;m second of X plus second squared off axe DX. And now let&#x27;s split the through into two separate smaller interests of pi times sort of pirate three. And we have science critics. And the usual way we do is remember I using the double angle formulas. So this is one Meyer&#x27;s car sign are you? Course I knw off to X. So I brought two DX and in the middle we have two pi times will sink and this one are co sign. So this is simply zero to pi over three and intentions ofthe X t X and plus pine off, sir, the pyre worth three second squirt of X t x and let&#x27;s continue. Let&#x27;s give Great. So then do it tough. This guy here, this pie we have one over to which is so it&#x27;s X over too. Linus and Heather it, of course, on sign with to you they&#x27;re gonna sign on to excel or and that zero and Pirates three and then plus on the next page because to party times. And we know that the editor with their attention this natural art of co sign. So it&#x27;s negative. Natural art off outside volume of co sign of acts evaluated against zero and pi three. Yes, and also we know that editor there Second square simply tangent So it&#x27;s we have high times high times tangent X a point sirrah too I overthrew so the 1st 1 c at Mexico&#x27;s zero. We have both syrup I or three we have buy or three minors. Pine tires, pirate three miners sign off to pie over three, divided by four and sign of two pi over three years. That sign off 60 degrees and there would be, I swear they&#x27;re three divided by two. And so they will be so with the two will have. For there is Rhys Forces Paige. There was no sex or force, so they will be, um, six here and the 1st 2 pi times. What&#x27;s happening here? We have natural log off co sign of fireworks free, that is course under 60 degrees. That&#x27;s 1/2 and then we have natural. Of course, I&#x27;m zero. It&#x27;s course I won that zero. So we&#x27;re gonna have We&#x27;re going to have and see Yes. So that&#x27;s going to be a natives. Negative natural. Log off one out of here and plus finally pi times tension of pie or three that sign of pie or three if I buy Coast side of pirate through that squared of three. So if I if we take everything out that say that steak pie out here, Big Prentice&#x27;s and they have pie over six Script of three divided by eight plus two times. Well, here we have natural log of one over to with their sign. It&#x27;s plus sign that short of to simply and square there. Three. So it looks like it&#x27;s going to be a pie squared over six plus two pi natural log, too, Plus, um, seven over eight square there three times pi.

J Hardin · Answer

here. We like to find the volume obtained by rotating the region bounded by the curves C can co sign for X between zero and pi over three around the line y equals negative one. So over here on the right, there&#x27;s a rough sketch of the graphs and read up here. In the first question, he was co sign from excess from zero power three. So coastline starts off at one and then co signed off. However, three&#x27;s a half, and here in blue, we have seeking. Speaking of zero was one and then c can of pirate three is too. Then we have the line y equals negative one, and we reflect this region between the curves, the blue and the red around the line y equals negative one. So here&#x27;s a recollection down here in the fourth quarter, so red is the reflection of the coastline. Blue is the reflection of the sea can, and here we see, we have a cross section, which is a washer. So this tells us that the volume will be of the form pie times of the large radius outer radius of the washer, minus the owner or the smaller radius the washing and the reason will use the X is because the volume is obtained by letting the washer moving the ex direction from zero to five for three. So next we have to find expressions for capital R, and little are so let&#x27;s start off with upper case are so coming over here is our picture We see ours the outer radius. So here&#x27;s the center of the washer and red right here and then we obtain the outer radius by going from the centre all the way up to the outer edge of the washer. So this is our upper case are and we see that this lower portion, this first part of our down here I&#x27;m scribbling in blue. That&#x27;s just the distance from negative one zero. So that starts off. There&#x27;s just one. And then this upper region up here, it&#x27;s just a Y value because it&#x27;s just from zero to a graph percents. This why value is on the blue graph it seek and so we could replace why with seek Innovex and similarly we could find a little R. So let&#x27;s label little are on the on the washer, so we start off from the center of the washer. And we go until the first curve that we hit, which is right there and read. So this is little R. So this first part of that are down here in the bottom below the access excesses scribbled in blue, that&#x27;s just one. And then we go from zero to this y value on the group. But this time we&#x27;re on the red curve, which is close on so that that those are expressions for the the radio guy so we could go ahead and plug goes in and then find the So we have You couldn&#x27;t pull that pie. Okay, that was good. And evaluate these squares Well, pie in a girl sort of laboratory. Then we have one plus two thinking. Plus, he can square minus one minus two co sign minus co sign squared, see if we could cancel in here. We see those ones, cancels, Let&#x27;s cross those off. Also, we can rewrite this co science. Where is one plus two co sign of two eggs over too s. So this is the double angle formula for co sign, and it&#x27;LL make it easier to generate the other terms or we already know the intervals of sea can of seek and squared and co sign but co sign squared will be easier to integrate after replacing it with this term over here. So let&#x27;s go to a new page. We need more room. So picking up where we left off Pi integral Part three. Then we have sequence where plus to see again. Minus one over two minus coastline to x over too minus two co signed X So let&#x27;s go ahead and integrate each of these. We know the integral of sequence where this tangent in a girl of Sikkim is. So here we have the two and then we have natural log Absolute value. C can&#x27;t plus tangent. Then for one half becomes excellent too. And for the coastline to X, we have signed to X over two times two, which is for so again here. If this two exes throwing you off, you could go ahead and use a U substitution. New equals two X And then finally this negative to co sign becomes a negative to sign X and the end points are zero pirate Drea. So it&#x27;s good and plug in these points. So tangent of Pirate three&#x27;s Route three so grew three plus two times natural log, absolute value. Seeking a power three is too. So we have two plus radical theory and then minus pi over three, divided by two since Piper six and then we have a minus sign of two pirate three. So that room three over too, divided by four. So that would be rude. Three over eight and then minus two times, sign a pie over three fifty times. Room three over too. So this is after plugging in Pirate three. And then when we plug in zero, we have tangent zero zero with two natural log seeking a zero was one tangent zero zero minus zero over too. And then sign of zero zero. So we have minus zero minus zero and we know that natural other born zero. So all these terms are zero. So now we just simplify the expression that we have also notice Over here these will cancel. So it&#x27;s combined these radical theories. So doing so, Combining those fractions we should get negative Route three over eight. We still have this minus five or six and then plus to natural log two plus route three and we could drop the absolute value here because two plus room three is positive. So no need for absolute values here. So this is our final answer, and that&#x27;s the volume.

Problem

Solve the given problems by integration. Find the…

Problem 38 Easy Difficulty

Solve the given problems by integration.
Find the volume generated by revolving the region bounded by $y=\tan ^{3}\left(x^{2}\right), y=0,$ and $x=\frac{\pi}{4}$ about the $y$ -axis.

Answer

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus

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

Heather Z.

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus