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IIl du3:00 PM83%webworkaurakac.aeHW-I: Problem 10Previous ProblemProblem ListNext Problempoint) Find the volume of the solid formed by rotating the region enclosed ...

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IIl du3:00 PM83%webworkaurakac.aeHW-I: Problem 10Previous ProblemProblem ListNext Problempoint) Find the volume of the solid formed by rotating the region enclosed byx =0,x =l,y=0 y=9+x6about the X-axis.37/4(pi)

IIl du 3:00 PM 83% webworkaurakac.ae HW-I: Problem 10 Previous Problem Problem List Next Problem point) Find the volume of the solid formed by rotating the region enclosed by x =0,x =l,y=0 y=9+x6 about the X-axis. 37/4(pi)



Answers

Problems $44-49$ concern the region bounded by $y=x^{2}$, $y=1,$ and the $y$ -axis, for $x \geq 0 .$ Find the volume of the solid. The solid obtained by rotating the region around the $y$ axis.

Alright we're gonna draw a picture then we're gonna send that region around the X. Axis and find its volume. Okay what was the secret of X. She hi um Like better make some room here. All right this will be one. Okay so seeking X. Is the reciprocal of the co sign? And the coastline goes through 10 -101. So wherever the coastline is zero the second has an assistant coach. So here and here And if the co sign is one the sequence is one or minus one. If the co sign is positive the seeking is positive. If the coastline is zero then the second goes to positive or negative infinity. Okay. Between zero and Pi or to the coastline is positive. So this part of the secret goes up like this and it goes down like this then it goes up like this that's the second And then the line is one Plus three Pi over X. Okay so it's why intercept is juan and its slope is up three over pie. Okay. Okay. Yeah. Okay. That wasn't so bad. So here's the area that we're talking about. So let me erase all this other stuff. We don't need all this. Okay. Goodbye drawing. Okay now we're gonna send that around the X. Axis. Okay? None of no piece of that picture is touching the axis of rotation. So I'm going to get a hole in my shape here. My disk. I'm going to slice this way. So the top the green line will make the outside of the disk. There it goes. And then the blue one that the second will make the whole. Okay. Okay so the volume will be pie big R squared minus little R squared times age can h stands for the thickness of the slice which you can see is a little bit of X. So dx big R. Is the distance from the axis of rotation to the outside of the disk. So from here to the line. So I'm going to call that Y. On the top which was the line one plus three over pi X. And then little R. Is the distance from the axis of rotation to the whole. So from here to here and that's why on the bottom and by the bottom, I mean the bottom of this area that we're sending around and that was the second of X. Okay now we're gonna add them up These little slices starting here at X. equals zero and stopping here at this point where those two things meet. So remember the first one was seeking ex and we need to know where it hits one plus three pi over X. True. Well it's pretty close to pi over two. Uh I don't know how to do this. Let's see. Um. Hi. Okay. Okay lets assume let's assume that it's one that we know. Okay so the ones that we know are 45 30 and 60. So let's let's work on the 30 and 61st. Okay. Here will be 60 which is five or three. That will be square 2321. Okay. The second of pi over three is hi pot news over adjacent. Oh I got it the first time one plus three by 543 one plus one is two. Okay. Okay so the volume is gonna be pi From 0 to Pi over three. Big are square. Oops I forgot the X minus little are square D. X. Can I gotta square that out? So I get one plus three pa three hour pi plus three of her pie. So six our pie plus three. Open six over pi X plus three over pi X quantity squared so nine pi squared X squared minus second school. Hi integral guan is X plus six over pi X squared over two Plus nine over Pi squared X cubed over three minus the integral of the second square. Well the second squared is the derivative of the tangent. So it's anti derivative. Its tangent. zero Pi over three. Hi Pi over three plus. Okay right here I'm gonna cancel the two with the six and have a three. So three over pi times. Hi over three square. Which I'm going to write his five or 35 or three plus. I'm gonna do the same thing here. This three will cancel into the +93. So I got three over pi pi Times, pi over three cubed pi pi by 333 minus the tangent of pyro three. Um tangent is opposite over adjacent. So square root of 3/1 minus zero plus zero plus zero minus the tangent of zero which is zero. Okay. Yeah fully. Can this 11 cancel? So I got fired for three. That's this one. Hi over three plus here. The pies. Two of the pies cancelled one of the threes cancelled over nine minus the square root of three. Hi, this is three pi over 93 pi over 91 pie over nine. So seven pi over nine minus the square root of three. Okay. That is weird. Okay. But it was a weird problem because it had trig functions and it had polynomial vials sort sort of uh with that weird three over pi thing in there. So I'm gonna say that's the answer.

All right, So this question wants us to revolve the area between these two curves around the line. Why equals 15? So let's do a rough sketch. We have our and then we have a line over here, you know, once here in the first quadrant. So this is our region, and we're evolving and around. Why equals 15 which, for simplicity's sake, let's just put that up here. It's not to scale. So since this line and region is not touching, why equals 15? We have to use the washer method. So first, let's draw outer idiots. So we see here that are, oh equals top minus bottom 15 minus the red curve, which is X squared and then are inter radius equals top minus the bottom, which in this case, is the line. So no, we confined our volume. Volume equals the integral from zero to our intersection point, which we confined real quick by setting the two equal to each other. X squared equals 12 minus acts. Then collect all of it one side product of negative 12 some of one that would be X plus for times X minus three. So X equals negative for and X equals positive. Three. So which one of those do we want? Well, this intersection is in the positive X. So we want positive three Integral from 0 to 3 of our outer radius squared, which is X squared. No, just 15 minus x square. Quantity squared minus our inner radius squared, which is 15 minus 12 minus x squared. This is all district acts and that simplifies Thio 1953 over five. Hi.

This problem wants you to find the volume of the solid form by rotating the shaded figure around the X axis. So if I rotate this over the X axis this was the X. This was the y axis. I would end up getting a three D. Hone shape. Kind of hard to drop but we have a cone here. The radius would be too and its height would be three from here to here would be three. So let's calculate the volume. Remember the volume is equal to one third area, the base times the height. So in this case volume would be equal to 1/3 area the base It's a circle with a radius of two. So that would be pie times two squared by r squared times up by the height of three. What this works out to then one third times three essentially reduces down to one. So you get a volume of four pi for pie. Um units squared r units cubed. Except gives me so or that volume. If I multiply four multiplied by high, I got approximately 12.5 12.5 seven units cubed.

They give the question some. The topic electronic contribution and predictable in this question is asked that Consider this set of 400 numbers for electrons and atoms. So there are four sets of data given and I've been asked to find the answer for the following questions. The first one is identified. The two interns that have the same spin identify. The two returns are in the same atomic orbiters. Now from the concept that we know about quantum numbers we know that the end is the significance of shell and it's also called US principle quantum number. It signifies shell partial number of an ethnic positive. L. It's called the metal content number. Let's call it as a metal quantum number. And it signifies me section. It signifies the social enamel signifies magnetic quantum number, magnet a condom number. And how about and it's the significance of orbital. It signifies which orbital the electron is in and the last one M. S. Are also called US. The spin content work gives an idea about what do the spin that electron is having right now. So M. S. The significance of the spin that I return has. So now let's look at the question spent quantum number which has an idea about this bit of electron. So let's have an idea and look at the question to solve this. So first one part of the question says I never do the letter that have since been. So spin is to remind my name is been is determined by they remained by the value of this. So M. S. Is same for data set three and four Observe it has three and ford has and this value as minus up. So they have the boat have same minus half US values. Therefore that would do the same one. The answer is 2, 3 and four. Later. The big questions is and it will turn that are in the same dwelling arbiters. A chemical releases undermined by atomic orbital. Don't remind by remind by as we can see here. Mm Ayman. So the mm value is same for the mean value here is same for since they're asking. So it must be of the same shells. It has to be I'm told that in the letter in the same atomic orbital. So it is about german M L and L. Its subsequent they should be seen. So first of all there and it should be seen and then there should also be seen. You can clearly see here the data said given the question here is the rule. So it has to be first of all, data. Number two is totally wrong because this cannot be zero on data. Number one can be correct. Basically it has to be one and 3 because this tour totally wrong. Option two and option number four um Are totally so it has to be one and 3. So because Ml the same. So that's 31 and three because they have same level and S. M. L. And L. Both helps them MLL and which is really correspond and really correspond. That's where the answer should be one and 2. So based on this, the answer for this question will be basically the answer. The question will be for Finnish reminder developments That there should be three and 4. And and we go over to listen to the man behind and then Just saving gets off one and 3. We gotta chosen 24 also has an answer. But the thing is that is wrong because that image that you cannot veto and develop can never be equal to Can never be more than in it. That's why option number two and optional food. It doesn't seem to be the correct one. The only answer


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