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# Disc method around y-axis

AP.CALC:
CHA‑5 (EU)
,
CHA‑5.C (LO)
,
CHA‑5.C.1 (EK)

## Video transcript

here we have the graph or part of the graph of y is equal to x squared again and I want to find the volume of another solid of revolution but instead of rotating around the x-axis this time I want to rotate around the y-axis and instead of going between 0 and some point I'm going to go between Y is equal to 1 and Y is equal to 4 so I'm going to do is I'm going to take this graph right over here I'm going to take this curve instead of rotating it around the x-axis like we did in the last few videos I'm going to rotate it around the y-axis so I'm going to rotate it around I'm going to rotate it around just like that so what's the shape that we would get let me see if we can visualize this so the base is going to look something like it's going to look something like that we could see through it look something like that and then this up here the top of it would look something would look something like that it looks something like that and we care about the stuff in between so we care about this part right over here not the very bottom of it and let me shade it in a little bit so it would look something it would look something like that so let me draw it separately just so we can visualize it so I'll draw it at different angles so if I were to draw it the y-axis kind of coming out the back it would look something like this it looks something like it's a little bit smaller like that and then it gets cut off right over here right over here like that so it looks I don't know what what's shape you could call it but I think hopefully you're conceptualizing this let me do that same yellow color the visual is in that's not yellow the visualization here is probably probably the hardest part but as we can see it's not too bad so it looks something like looks something like this looks like maybe a truffle an upside down truffle so this writer and let me draw the y-axis just just so we're oriented so the y-axis is popping out is popping out in this example like that then it goes down over here then the x axis the x axis is going is going like this so I just tilted this over I tilted it over a little bit to be a view it at a different angle this top right over here this top right over here is this top right over there so that gives you an idea of what it looks like but we still haven't thought about how do we actually find the volume of this thing well what we can do instead of creating disks where the depth is in little DX is what if we create a disks where the depth is in dy so let's think about that a little bit so let's create let's think about constructing a disk at a certain Y value so let's go let's think about a certain Y value we're going to construct a disk right over there that has the same radius of the shape at that point so that's our disk that's our disk right over here that's our disk right over there and then it has a depth instead of saying it has a depth of DX let's say it has a depth of dy so this depth right over here is d Y so what is the volume of this disk in terms of wine as you can imagine we're going to do this definite integral it is a definite integral with respect to Y so what's the air what's the volume of this thing well like we did in the last video we have to figure out the area of the top of each of these disks or I guess you could say the face of this coin well to find that area it's PI R squared if we can figure out this radius right over here we know the area so what's that radius so to think about that radius in terms of Y we just have to solve this this this equation explicitly in terms of Y so instead of saying it's y is equal to x squared we can take the principal root of both sides and we could say that the square root of Y is equal to X and this right over here is only defined for non-negative Y's but that's ok because we are in the positive x axis right over here so we can also call this function we can also call this function right over here X is equal to the square root of Y and we're essentially looking at this side of it we're not looking at we're not looking at this stuff right over here so we're only looking at this side right over here we've now expressed this graph this curve as X as function of wise if we do it that way what's our radius right over here well our radius right over here is going to be f of Y it's going to be the square root of Y it's going to be the square root of Y is our radius so it's going to be a function of Y I don't want to confuse you for if you thought this was f of X and I think this is f of Y I know it would be a function of Y we could call it G of Y it's going to be the square root of Y so area area is equal to PI R squared which means that the area of this thing the area of this thing is going to be pi times our radius times our radius squared our radius is square root of Y our radius is square root of Y so this thing is going to be equal to it is going to be equal to is going to be equal to PI square root of Y squared is just pi times y now if we want the volume we just have to multiply the area of the surface times the depth times dy so the volume of each of those disks is going to be PI Y times dy x times dy this gives you the volume of the disk volume of a disk now if we want the volume of this entire thing we just have to sum all of these disks for all the Y values between Y is equal to 1 and Y is equal to 4 so let's do that so we just take the definite integral from y is equal to 1 and y equals 4 and just as a reminder definite integral is a very special type of sum we're summing up all of these things but we're taking the limit of that sum as these d wise gets get shorter gets I guess squatter and squat or smaller and smaller and we have a larger and larger number of these disks really as these become infinitely small and we have an infinite number of discs so that our sum doesn't just approximate the volume it actually is the volume at the limit so to figure out the volume of this entire thing we just have to evaluate this definite integral in terms of Y and so how do we do that well it's going to be equal to what's we could take the PI outside it's going to be PI times the antiderivative of Y which is just Y squared over 2 - y squared over 2 evaluated from 1 to 4 evaluated from 1 to 4 which is equal to pi times well you evaluate it for you get 16 over 2 let me just write it out like this 4 squared over 2 minus 1 squared over 2 which is equal to pi times 16 over 2 is 8 minus 1/2 and so we could view this as 16 halves minus 1/2 which is equal to 15 halves so this is equal to 15 halves 15 halves times pi or another way of thinking about it 7 and 1/2 times 5 this is a little bit clearer so we're done we found our volume not rotating around the x axis but rotating around the y axis which is kind of kind of exciting
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