hello calp kids this is mr bean welcome back to another lesson in calculus and today we're going to continue on working with volumes of a solid of revolution where we revolve things around and we're going to stick with washer method and as a reminder what's washer method that's when you have a cross section that's a circle with a gap in the middle so you have if you you take the solid and you cut it up slice it down the middle you have this area on the outside but then there's an opening in the middle so this is what we did last time we took pi r squared minus little r squared we had the big radius squared minus the little radius squared and then of course the pi makes it the area of each circle so this is what we did our in our last lesson so if you have no idea what i'm talking about then you're watching the wrong lesson you got to go back one to be able to follow what we're doing today we're going to take these revolutions and instead of doing it around the x or y axis down here we are now going to do it around a new line so on this first problem let's do it around the line y equals two so draw yourself like a dashed line at y equals 2 and we're revolving this thing this direction that's how we're going to be revolving it so let's take our area that's enclosed by the square root of x and x squared and we need to create a mirror image so let's come up here and try to create a mirror image of this thing what's that going to look like so an upside down parabola would be here oh and it's all the way it's two away so i need to start two way here so something like this there's my upside down parabola and then there's the square root going the other other direction so there is my mirror image that's not great but it's good enough for what we're trying to do and then let's revolve it so how do we revolve it we create like these little circles oval shape type thing so do the best you can with that and show that you're revolving it both directions and then the right side here you do that here as well hopefully you can visualize what in the world this is doing it's a very uh it's a big shape that goes around a circle with a huge hole in the middle here see this whole thing is empty on both sides it would be just open up you could look straight through it so that's our volume now let's figure out how to set up our equations i'm going to draw a line like this i would like you to as well because i'm going to show you two different directions two different ways of setting this up so the first way is going to be how most textbooks will do it uh and i'm going to show you a second way that i believe is actually easier so let's do start off with our pi and then what's the low x value we're going from zero to one so we're taking focusing in on this area that we started with so we're going from zero to one and then we have the big radius squared minus the little radius squared with respect to x do not forget the pi do not forget the pie we have a pie right there it needs to be in front that's a common mistake you start setting these things up with the radius squared and you forgot oh yeah circles pi r squared not just an r squared so we start with the big radius so the big radius is starting right here in the middle and you go to the outside of the object which is down here to that point so that's my big r so what is that distance well i know that this distance right here is what is that that's the x squared right that distance is x squared well if that distance is x squared then the this thing in red that is the whole thing which is 2 minus x squared so that's how you get the big radius now what about the little radius the little radius we'll let's draw this again i'm going to go from there to there that was not a straight line we're pretending like it is so i need that distance again i don't know that distance but i do know the whole thing the whole thing is 2 so i could take 2 and then subtract this down here what is that that is the square root of x right there so i subtract the square root of x and there's my setup so that's finding the volume and again yes we'd have to we'd probably use a calculator on this one because that's a lot of multiplying out just depends on what the problem is so let me show you a different way of doing this that may be a bit easier for you i do not want to confuse you so you can do whichever way makes sense and if what i just told you makes total sense then maybe you don't even care to watch this part but i'm telling you for most kids this is actually easier so we're gonna still set this up the same way we're doing it from zero to one but we're going to now we i've done this before we're going to now instead of saying that this is the line y equals 2 we're going to shift it down 2. i'm going to subtract 2 all the way down and so that this now becomes the x axis so shift everything down 2 units so that means my new equation here is going to be y equals the square root of x minus 2 because i shifted everything down and this one is now y equals x squared minus 2 because i shifted everything down now i can do it exactly like we did in the last lesson these two equations we shifted it down two so it's now the x-axis and we just start from there so think of this as the x-axis what is it to the outside to that outside is that graph and that graph is x squared minus 2. so i'm going to write the big radius squared minus the little radius squared with respect to x and the big radius squared is this line right here right that line so that one is x squared minus two and then we subtract the little radius and the little radius show you in here is to that line so we're going from the x axis to that square root curve and the square root is square of x minus two so i say square root of x minus 2. now if you look at these they're different right those two answers are different we have 2 minus x squared quantity squared and here we have x squared minus 2 quantity squared but if you remember a couple lessons ago i taught this about i reminded you that if you have 5 minus 3 and square it that's exactly the same thing as 3 minus 5 squared when you square something the sign won't matter it will always become positive so both of these will give you the correct answer so you're allowed to either just vertically shift it or just look at it right from the line y equals two and kind of figure out what is that distance how to get the distance of the whole thing minus what you didn't use all right let's do one more of these this time it'll be a vertical line x equals negative one let's go ahead and draw a vertical line at x equals negative one i like to keep them dashed to remind me that it's actually not part of the graph it just helps me know what it is and we're doing this area and then since we're doing it around the x uh x equals negative one line that's kind of similar to doing it around a y axis it's a vertical line that we're revolving around so we need to have these equation in terms of y so let's solve this one if in terms of y i'd get y minus 1 equals x squared square root both sides you get y minus 1 equals x and i could take plus or minus except that i already know that this is the positive side so i don't need the plus or minus i just need the positive square root of y minus 1. so that's this curved line and now the straight line here if i subtract 1 y minus 1 and then divide by 2 i get this fraction or you could write it as y over 2 minus 1 half alright so now there's my two equations i'm going to draw a line here to show you how to set up both versions oh i better let's finish drawing my graph i got to do the mirror image over here so let me fast forward to that all right not perfect but good enough and then we're going to have you revolve it around this direction and oh that was bad you revolve it like that and then well there's a big circle this is hard that's why i do little tick marks because then it's more forgiving when i mess up i can just kind of scroll down okay okay yeah yeah yeah so big hole on this one as well when we're revolving it around this line x equals negative one so let's start this off what do we got going here we've got that the volume is going to be pi r squared for the area of a circle and then we're integrating it from remember this is with respect to y so we start at the bottom which is a y equals one value one and then we go to the top of it and the top is a y value of five and then i'm going to have the big radius squared minus the little radius squared all with respect to y so the big radius is right here from this line to the outside of the object to right there so what is that well i know from here from the the y-axis to the curve that's this square root of y minus one but this part right here that distance is 1 so it's 1 plus the square root of y minus 1. so we go 1 plus the square root of y minus 1. and then the other one i'll do this one in blue that distance right there is one plus and then this is that weird fraction there y minus one over two so we say one plus y minus one over two and that's we're subtracting that then we have a minus that we're subtracting this because that's the little radius where there is a gap if there's a hole in this three-dimensional object okay so that's the volume that's how you set it up and you can use a calculator to evaluate it from there let me just show you the other direction where if you just shift everything to get exactly the same answer so we go from one to five and now all i'm doing is i'm going to shift this whole thing to the right one so i'm going to add one to everything so my new equations i'm going to write it ah you're just doing this you're just taking a plus one to that and then for this one you're doing a plus one that's it you just take these equations and add one to them that's all you have to do so now that this is the x-axis you have to think this is actually the x-axis now and these curves are these things with a plus one on them you don't have you've already shifted it you don't have to worry about any type of other line because it's now the x axis so we say big r squared minus little r squared with respect to y and so now what's the big radius from the from the what i said y axis i mean i said x axis i'm at y axis sorry from the y axis to the outside uh that is the curve one this one square root of y minus one square root of y minus one plus one because i added one for shifting it and then the the line is y minus one over two plus one because i added one because i shifted it and then that's it there's there is your setup and you can see it's actually the same thing as the one up above so whichever one you're more comfortable with just shifting it right off the bat shift it back to the y-axis or the x-axis and then work from there but just remember you have to add or subtract on to the equations that you already came up with all right now this entire lesson with this packet you're going to have a lot of review things including on the master checks you'll have review from other things of unit 8 you've got to find the area you got to do perpendicular cross sections just to kind of keep it straight because that's one of the hardest things to do is remembering when you use each of these different formulas that we've come up with so you really got to understand that well so where i'm throwing all of it out at you at this lesson and that's actually a really good way to review before you're getting ready for the test so to my bc kids you're done rock the master check bc kids i'll see you back in the next lesson you got one more before you're done with unit eight a b kids stick around for just a minute i want to say some parting words to you and give you some recommendations of how to get ready for this ap exam so here's what i would tell you to do practice exams do as many as you can between now and the time you take the ap exam do as many practice exams as you can and time yourself stick to the time you're supposed to stick to and don't give yourself more time that's going to help you quite a bit my suggestion if you don't have a way of getting any your hands on any practice exams you may go online and buy a barren's ap review book for calculus like a cram book or princeton review those are the two that are my favorite but your teachers might have some other suggestions or maybe you already have some so those have really good tests in the back of them use those time yourself that's a great way to prepare then you take those exams and correct your errors go through figure out what you missed and what is going on and why you miss them as you do that don't get stressed out about the ones that are really hard if you have a chrome across ones that are really hard you know you have to think about what's your goal here you're not going to get 100 and the reason i can say that is because last year there was only two kids who got 100 percent okay two kids out of hundreds of thousands of kids who took the ap exam the cal kp exam only two got a hundred don't worry about getting a hundred okay you don't need that it's like i think it's like 65 or something like that is about is about a five so you don't need to be stressed about getting them all right but it's great it's great to go over them but if you're struggling this year don't stress about the real hard ones okay you just make sure you're getting about half of them right and you're gonna probably end up with a four and you're doing pretty well um still spend some time trying to understand them but if they're really hard don't get stressed about it that's also why you're taking it too when you're taking the practice exam do not slow down on really hard ones that you don't think you're gonna get if you know that they're beyond your abilities just go past it and keep moving okay and then procedural things you've got to be automatic and what i mean by procedural is that if you have a function that says f of x equals the square root of sine x you have to know how to take the derivative of that okay you gotta know if you don't know how to do the chain rule and taking the derivative you're in trouble procedurals have got to be automatic or if i say let's take the integral of sine of 3x if you don't know how to take the integral of sine of 3x by using u substitution then you're going to be in trouble you've got to know those types of things and they got to be automatic okay all right look that's it we have finished the whole year this is fantastic isn't it a relief feels pretty good to be done so uh not only are you gonna rock that master check you're gonna rock the ap exam the year's been fun i'm glad we could spend some time together with you even though most of you who are watching this i don't even know who you are but this is mr bean signing off i'm out maybe for those of you who are bc then you're still listening i don't know why you're still on this video but i'll see you guys next lesson a b kids that's it you