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Main content
Current time:0:00Total duration:8:31
AP.CALC:
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Video transcript

let's now generalize what we did in the last video so if this is my y-axis and this is that's not that straight this right over here is my x-axis and let's say I have two functions so I'm just going to send in general terms so let's say I have a function right over here so let's say it looks something like this so that's one function so this is y is equal to f of X and then I have another function that Y is equal to G of X so let's say it looks something like this so Y the blue one right over here is y is equal to G of X and like we did in the last video I want to think about the volume of the solid of revolution we will get if we essentially rotate the area between these two and if we were to rotate it around the x axis so we're saying in very general terms this could be anything but in the way we've drawn it now it would literally be that same truffle shape it'd be a very similar truffle shape where on the outside it looks like a truffle on the outside it looks like something like a truffle and on the inside we have carved out a cone obviously this visualization is very specific to the way I've drawn these functions but what we want to do is generalize at least the mathematics of it so how do we find a volume well we could think of disks but instead of thinking of disk we're going to think about washers now which is essentially the exact same thing we did in the last video mathematically but it's a slightly different way of conceptualizing it so imagine imagine a imagine taking a little chunk between these two between these two functions just like that what is going to be what is going to be the width of this chunk well it's going to be equal to DX and let's rotate that whole thing around the x-axis so if we rotate this thing around the x-axis we end up with a washer that's why we're going to call this the washer method it's really just kind of the disk method where you're where you're gutting out the inside of a disk so that's the inside of our washer and then this is the outside of our washer outside of our washer or look something like look something like that hopefully that makes that makes sense and so the surface of our washer the surface of our washer looks something like that I know I could have drawn this a little bit better but hopefully it serves the purpose so that we understand it so the surface of our washer looks something like that and it has depth of DX it has depth D X so let me see how well I could draw this so depth D X that's the side of this washer so a washer you can imagine is kind of a gut it out it's a gutted out coin it's a gutted out coin so how do we find the volume well once again if we know the surface if we know the area of the face of this washer we can just multiply that times the depth so it's going to be the area of the face of the washer so the area of the face of the washer well it could be the Varia if you had if it wasn't to got it out and what would that area be well it would be pi times the overall the outside radius squared it would be pi times the outside radius squared well what's the outside radius the radius that goes to the outside of the washer well that's f of X so it's going to be f of X is the radius and we're just going to square that so that would give us this expression right over here would give us the area of the entire face if it wasn't a washer if it was a coin but now we have to subtract out the inside so what's the area of the inside what's the area of the inside this part right over here well we're going to subtract it out it's going to be pi times the radius of the inside squared the radius of the inside squared well what's the radius of the inside squared well the inside in this case is G of X it's going to be pi times G of x squared that's the inner function at least over the interval that we care about so the area of this washer we could just leave it like this or we could factor out a PI we could say it's equal to the area is equal to if we factor out of Pi pi times f of x squared f of x squared minus G of x squared minus I have to write a parenthesis there so correct f of x squared minus G of X squared and then if we want the volume for that in the same yellow if we want the volume of this thing we just multiply it times the depth of each of those washers so the volume of each of these washers are going to be pi times pi times f of X f of x squared minus G of x squared the outer function squared over our interval minus our inner function squared over the interval and then times our depth times our depth that'd be the volume of each of these washers now for eat and that's going to be defined at a given X in our interval but for each exit these interval we can define a new washer so there could be a washer out here and a washer out here is we're going to take the sum of all of those washers and take the limit as we have smaller and smaller depths and we have an infinite number of infinitely thin washers so we're going to take the integral over our interval from where these two things intersect the interval that we care about it doesn't have to be where they intersect but in this case that's what we'll do so let's say x equals a to x equals B although it could have been from this could have been a that could have been B but this is our interval we're saying in general terms from A to B and this will give our volume this right over here is the volume of each washer and then we're summing up all of the washers and taking the limit is we have an infinite number of them so let's see if we apply this to the example in the last video whether we get the exact same answer well in the last video y equals G of X was equal to X and y is equal to f of X was equal to the square root of x so let's evaluate that given what we just were able to derive so our volume ended up here the volume is going to be the integral what are the two intersection points well over here once again we could have defined the interval someplace else like between there and there and we would have got in a different shape but the points that we care about the way we visualize it is between X is equal to 0 x is equal to 0 and X is equal to 1 that's where these two things intersect we saw that in the last of pi pi times what's f of x squared f of x squared square root of x squared is just X minus G of x squared G of X is X that squared is x squared and then we multiply times DX so this is going to be equal to we can factor out the PI 0 to 1 X minus x squared DX which is equal to PI times let's see the antiderivative of X is x squared over 2 antiderivative of x squared is x squared over 3 X sorry X to the third over 3 and we're going to evaluate this from 0 to 1 so this is going to be equal to I'm running out of my real estate a little bit let me let me scroll over to the right a little bit so this is going to be equal to PI times well if when you evaluate this whole thing at 1 you get see you get 1/2 minus 1/3 1/2 minus 1/3 and then you subtract it evaluated at 0 but that's just going to be 0 0 squared over 2 minus 0 third over 0 to the third PI over 3 that's just all going to be 0 so when you subtract out 0 you just left with this expression right over here what's 1/2 minus 1/3 well that's 1/6 this is 1/6 and so we are left with this is equal to PI over 6 which is the exact same thing that we got in the last video and that's because we did the exact same thing that we did in the last video we just conceptualized it a little bit differently we generalized it in terms of f of X and G of X and we essentially conceptualize it as we conceptualized it as a washer as opposed to kind of doing the disk method for an outer shape and an inner shape like we did in the last video