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Current time:0:00Total duration:8:03

Video transcript

in the last video we figured out the volume between this surface which was XY squared and the XY plane when X was went from 0 to 2 and Y went from 0 to 1 and with the way we did it is we integrated with respect to X first we said let us hold Y pick a Y and let's just figure out the area under the curve and so we integrated with respect to X first and then we integrated with respect to Y but we could have done it the other way around so let's let's do that and just make sure we got the right the right answer so let me erase a lot of this so remember our answer was 2/3 when we integrated with respect to X first and then with respect to Y but I will show you that we can integrate the other way around that's good when you can get the same answer in two different ways so let me redraw redraw that graph because I want to give you the intuition again so that's my x axis y axis z axis X Y Z and let me make then this is my XY plane down here alright Y goes from 0 to 1 X goes from 0 to 2 alright this is going to be this x equals 1 this is x equals 2 this is y equals 1 and then the graph I will do my best to draw it look something let me do it anymore and get some contrast going here so the graph looks something like this let me see if I can draw it this side it looks something like that and then it comes down comes down like that straight and then the volume we care about is actually this volume underneath the graph this is the top of the surface on that side and we care about this volume underneath the surface and then when we draw the bottom of the surface let me do it in a darker color like something like this this is the bottom underneath underneath the surface I could even shade it a little bit just to show you that it's like the underneath underneath the surface hopefully that's a decent rendering of it alright let's look back at what we had before right it's like a page that I just flipped up at this point we care about this volume kind of the colored area under there so let's figure out how to do last time we integrate with respect to X for us let's integrate with respect to Y first so let's hold X constant so if we hold X constant what we can do is for a given X let's pick an X and let's say what is so if we if we pick a given X I don't let's pick the X I know here then what we can do if we're given X you can view that function of x and y right if X is a constant let's see if you know if X is 1 then Z is just equal to Y squared right and that's easy to figure out the area under as we can see that X isn't a concept we can treat it as a constant so for example for any given X we would have a curve like this right and what we could do is we could try to figure out the area of this curve first so how do we do that well we just said we could kind of view this function up here as Z is equal to XY squared because that's exactly what it is but we're holding X constant we're treating it like a constant and to figure out that area we could take a dy a change in Y a dy multiply it by the height which is XY squared right so we take XY squared XY squared multiply it by dy and if we want this entire area we integrate it from Y is equal to 0 to Y is equal to 1 Y is equal to 0 from 0 to 1 fair enough now once we have that air if we want the volume underneath this entire surface what we could do is we can multiply this area times DX and get some depth to going pick a nice color that's green so let's say our that's our DX so if we multiply that times DX we would get some depth let me do a darker color get some contrast sometimes I feel like that guy who paints on PBS DX so now we have the volume of this you can kind of view the area of the curve times the DX so we have some depth here so it's x DX if we want to figure out the entire volume under this surface between the surface and the xy-plane given this constraint to our domain we just integrate from X is equal to 0 to 2 X is equal to Z from 0 to 2 alright so once we so let's think about it this area this area in green here that we started with that should be a function of X right we held X constant but depending on which X you pick this area is going to change right so when we evaluate this magenta inner integral with respect to Y we should get a function of X and then well your value the whole thing will get our volume so let's do it let's evaluate this inner integral hold X constant what's the antiderivative of Y squared is y to the third over 3 right so it's Y third over 3 the X is a constant right and we're going to evaluate that at 1 and a 0 and the outer integral is still with respect to X DX this is equal to let's see when you evaluate Y is equal to 1 you get 1 to the 3rd that's 1 so it's x over 3 minus when y is 0 then that's whole thing just becomes 0 right so this this this purple expression is just x over 3 x over 3 and then we still have the outside integral from 0 to 2 DX so given what X we have the area of this green surface that babbles out where we started given any given X that area now I wanted something with some contrast this area is x over 3 depending on which X you pick if X is 1 this area right here is 1/3 right but now we're going to integrate across the the entire surface and get our volume and like I said when you integrate it it's a function of X so let's do that and this is just plain old vanilla standard integral so what's the antiderivative of X it's x squared over 2 we have a 1/3 there so it equals x squared over 2 times 3 so x squared over 6 we're going to evaluate it at 2 and at 0 2 squared over 6 is 4 6 minus 0 over 6 which is equal to 0 equals 4 6 what is 4 6 well that's just the same thing as 2/3 so the volume under the surface is 2/3 and if you watch the previous video you will appreciate the fact that when we when we changed when we integrated the other way around when we did it with respect to X first and then Y we got the exact same answer so that the universe is in proper working order and I've surprisingly actually finished this video with extra time so for fun we can just spin this graph and just appreciate the fact that we have figured out the volume between this surface XY squared and the xy-plane pretty neat anyway I'll see you in the next video