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Current time:0:00Total duration:4:51

AP.CALC:

CHA‑5 (EU)

, CHA‑5.C (LO)

, CHA‑5.C.2 (EK)

let's do another example and this time we're going to rotate our function we're going to rotate it around a vertical line that is not the y-axis and if we do that so we're going to rotate Y is equal to x squared minus 1 or at least this part of it we're going to rotate it around the vertical line X is equal to negative 2 and if we do that we get this gum ball shape that looks something like this so what I want to do is I want to find the volume of this using the disk method so I want to do is construct some discs so construct some disks so that's one of the disks right over here it's going to have some depth and that depth is going to be dy it's going to be dy right over there and it's going to have some area on top of it that is a function of any given Y that I have so the volume of a given disc is going to be the area as a function of Y times the depth of the disk times dy and then we just have to integrate it over the interval that we care about and we're doing it all in terms of Y and in this case we're going to integrate from Y is equal to well this is going to hit this y-intercept right over here's y is equal to negative 1 and let's go all the way to Y is equal to let's say Y is equal to 3 y is equal to 3 right over here so from y equal negative 1 to y equals 3 so y equals negative 1/2 Y is equal to 3 and that's going to give us that's going to give us the volume of our upside-down gumdrop type looking thing so the key here is so that we can start evaluating the double integral is to just figure out what the area of each of these disks are as a function of Y and we know that area is just area as a function of Y is just going to be pi times radius as a function of Y squared so the real key is what is the radius of as a function of Y for any one of these Y's so what is the radius as a function of Y so let's think about that a little bit what is this curve well let's write it as a function of Y if you add 1 to both sides and I'm gonna swap sides so you'll get x squared is equal to y plus 1 I just added one side 1 to both sides and then swap sides and then you get X is equal to the principal root of the square root of y plus one so this we can write as X or we can even write it as f of Y if we want F of Y is equal to the square root of y plus one or we can say X is equal to F of F function of Y which is the square root of y plus one so what's the distance what's the distance here at any point well this distance this let me make it very clear so it's going to be our total distance in the horizontal direction so this first part as we're I'm going to do it in a different color you can't see so this part right over here is just going to be the value of the function it's going to give you an x value but then you have to add another two to go all the way over here so your entire radius as a function of Y your radius as a function of Y is going to be equal to the square root of y plus one this essentially will give you one of these X values when you're sitting on this curve it's X as a function of Y I'll give you one of these X values and then from that you add another two so plus 2 another way of thinking about it you get an x value here and from that x value from that x value subtract out X is equal to negative 2 and when you subtract X is equal to negative 2 you're adding 2 here but hopefully this makes intuitive sense this is the x value let me do this in a better color this right over here this distance right over here is the x value you get when you just evaluate the function of Y but then if you want it the full radius you have to go another two to go to the center of our axis of rotation once again if you just take a given Y right over there you evaluate the Y you get an x value that x value will just give you this distance if you want the full distance you have to subtract negative 2 from that x value which is essentially the same thing as adding 2 to get our full radius so our radius as a function of Y is this thing right over here so substituting back into this we can now write our definite integral for our volume the volume is going to be equal to the definite integral the definite integral from negative 1 to 3 of pi times our radius squared dy so I can write the pi here we've done this multiple times times radius squared so it's going to be square root of y plus 1 plus 2 squared that's our radius x times dy so we've set up the double the definite integral and now we just have to evaluate this thing and I'll save that for the next video and I encourage you to try this out on your own