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Current time:0:00Total duration:6:38

Washer method rotating around horizontal line (not x-axis), part 1

AP.CALC:
CHA‑5 (EU)
,
CHA‑5.C (LO)
,
CHA‑5.C.4 (EK)

Video transcript

now let's do a really interesting problem so I have y equals x and y is equal to x squared minus 2x right over here and we're going to rotate the region in between these two functions so that's this region right over here we're not going to rotate it just around the x axis we're going to rotate it around the horizontal line y equals 4 so we're going to rotate it around this and if we do that we'll get a shape that looks like this I drew it ahead of time just that I could draw it nicely and as you can see it looks like some type of a I don't know a vase with a hole at the bottom and so what we're going to do is attempt to do this using I guess you could call it the washer method which is a variant of the disk method so let's construct a washer so let's look at a given X so let's say an X right over here so let's say that we're at an X right over there and we're going to do is going to rotate this region we're going to give it some depth D X DX so that is d X so we're going to rotate this around the line y is equal to 4 so if we were to visualize it over here you have some depth you have some depth and when you rotate it around the inner radius is going to look like the inner radius of our washer is going to look something like that looks something like that and then the outer radius of our washer is going to is going to contour around x squared minus 2x so it's going to look something something my best attempt to draw it it's going to look something like that something like that and of course our washer is going to have some depth so let me draw the depth so it's going to have some depth DX so this is my best attempt at drawing some of that to depth so this is the depth of our washer and then just to make the face of the washer a little bit clearer let me do it in this green color so the face of the washer is going to be all of this business the all of this business is going to be the face of our washer so if we can figure out the volume of one of these washers for given X that we just have to sum up all of the washers for all of the X's in our interval so let's see if we can set up the integral and maybe and then in the next video we will just we'll just forge ahead and actually evaluate the integral so let's think about the volume of the washer to think about the volume of the wash we're really just to think about the area of the face of the washer so area of face but face in quotes it's going to be equal to what well it would be the area of the washer if it wasn't a washer if it was just a coin and then subtract out the area of the part that you're cutting out so the area of the of the washer if it didn't have a hole in the middle would just be pi times the outer radius squared outer radius squared it would be pi times this radius squared that we could call the outer radius and then if we since it's a washer we need to subtract out the area of this inner circle so minus pi times inner radius inner inner radius inner radius squared so we generally just have to figure out what the outer and inner radius or radii I should say are so let's think about it so our outer radius our outer radius is going to be equal to what well we can visualize it over here this is our outer radius which is also going to be equal to that right over there so that's the distance between y equals 4 and the function that's defining our outside the distance between y equals 4 and the function that is defining our outside so this is essentially this height right over here is going to be equal to 4 minus x squared minus 2x I'm just finding the distance or the height between these two functions so the outer radius is going to be 4 minus this minus x squared minus 2x which is just 4 minus x squared plus 2x now what is the inner radius inner inner radius what is that going to be well that's just going to be this distance that's just going to be the distance between y equals 4 and y equals x so this is going to be 4 minus X it's just going to be 4 minus X so if we wanted to find the area of the face of one of these washers for a given X it's going to be it's going to be and we can factor out we can factor out this pie it's going to be pi times the outer radius squared which is all of this business squared so it's going to be 4 minus x squared plus 2x squared minus PI times the inner is although we factored out the PI so minus the inner radius squared so minus 4 minus x squared so this will give us the surface the a surface air or the area of the surface or the face of one of these washers if we want the volume of one of those washers we then just have to multiply times the depth DX we just multiply times the depth DX and then if we want to actually find the volume of this entire figure then we just have to sum up all of these washers for each of our X's so let's do that so we're going to sum up the washers for each of our XS and take the limit as they approach 0 but we have to make sure we got our interval right so what are these what are these we care about the entire region between the points where they intersect so let's make sure we get our interval so to figure out our interval we just say well when does y equals at when does y equal X intersect Y equal x squared minus 2x so we just have to say when does X let me just in a different color we just have to think about when does X equal x squared minus 2x equal x squared minus 2x one of our two functions equal to each other which is equivalent to at if we just subtract X from both sides if we just subtract X from both sides we get when does x squared minus 3x equals 0 we can factor out an X on the right hand side so this is going to be when does X times X minus 3 equals 0 well if the product is equal to 0 at least one of these need to be equal to 0 so X could be equal to 0 or X minus 3 is equal to 0 so X is equal to 0 or X is equal to 3 so this is X is 0 and this right over here is X is equal to 3 so that gives us our interval we're going to go from x equals 0 to x equals 3 to get our volume in the next video we'll actually evaluate this integral