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Current time:0:00Total duration:9:50

Video transcript

you hopefully have a little intuition now on what a double integral is or how we go about figuring out the volume under surface so let's actually compute it I think it'll all become a lot more concrete so let's say I have the surface Z and it's a function of x and y and it equals X Y squared the surface in three-dimensional space and I want to know the volume between this surface and the xy-plane and the domain in the xy-plane that I care about is X is greater than or equal to zero and less than or equal to 2 and Y is greater than or equal to zero and less than or equal to 1 and let's see what that looks like just so we have a good visualization of it so I graphed it here and we can rotate around this is Z equals XY squared and this is the bounding box right X goes from 0 to 2 Y goes from 0 to 1 so we literally want this you can almost view at the volume well not almost exactly view it is the volume under this surface between this surface the top surface and the xy-plane and I'll rotate it around so you can get a little bit better sense of the actual volume let me rotate it now I should use the mouse for this so it's this space underneath here you could it's got like a it's like a makeshift shelter or something and I could rotate it a little bit so you can see whatever is under this between the two surfaces that's the volume whoops I've flipped it there you go so that's the volume that we care about let's figure out how to do it and we'll try to gather a little bit of the intuition as we go along so I'm going to draw a not as impressive version of that graph but I think it'll it'll do the job for now let me draw my axes that's my x-axis that's my y-axis that's my z-axis X Y Z and we're going X is going from zero to two let's say that's 2 y is going from 0 to 1 and we're taking so we're taking the volume above this rectangle and the xy-plane and then the surface I'm going to try my best to draw I'll draw it in a different color I'm looking at the picture at this end it looks something like this it looks something like this and then it has a straight line see if I can draw this surface going down like that and then if I was really good I could shade it it looks something like this or shade it the surface looks something like that and this right here is above this right like this is the the top the bottom left corner you can almost view it so this is let me say the yellow is the top of the surface that's the top of the surface and then this is under the surface right so we care about this volume underneath here let me draw the let me show you what the actual surface of this volume underneath here alright I think you get the idea so how do we do that well in the last in the last example we said well let's pick an arbitrary Y and for that Y let's figure out the area under the curve so if we if we fix some Y if we fix some one when you actually do the problem you have to think of it think about it in this much detail but I want to give you the intuition so if we pick just an arbitrary Y here so on that Y we could think of it if we have a fixed Y then the function of x and y you can almost view it as a as a a function of just X for that given Y and so we're kind of figuring out the value of this of the area under this curve right you could this should you should view this as kind of an up-down curve for a given Y so if we know Y we can we can figure out then for example if Y was fine this function would become Z equals 25 X right and then that's easy to figure out the the value of the curve under so we'll make the value under the curve as a function of Y we'll pretend like it's just a constant so let's do that so if we have a DX that's our change in X and then our height of each of our rectangles it's going to be a function it's going to be Z the height is Z which is a function of x and y so we can take the integral so the area of each of these is going to be our function XY squared I'll do it here because I'll run out of space X Y squared times the width which is DX and if we want the area of this slice for given Y we just integrate along the x-axis right we're going to integrate from X is equal to 0 to X is equal to 2 from X is equal to 0 to 2 fair enough now but we don't still want to figure out the area under under the curve at one slice for one particular Y we want to figure out the entire area of the curve so what we do is we say okay find the area under of this under the curve not the surface under this curve for a particular Y is this expression well what if I gave it a little bit of depth right if I multiplied this area times dy then it would give me a little bit of depth right we kind of have a three-dimensional slice of the volume that we care about I know it's hard to imagine let me bring that here so if I had a slice here this is what we just figured out the area of us of a slice and then I'm multiplying it by dy to give it a little bit of depth so you multiply it by dy to give it a little bit of depth and then if we want the entire volume under the curve we add all the dys together take the infinite sum of these infinitely small volumes really now and so we will integrate from Y is equal to 0 to Y is equal to 1 I know this graph is a little hard to understand but you might want to re-watch the other the first video I the easier to understand surface so now how do we evaluate this well like we said that we you evaluate from the inside and go outward and it's almost like taking its it's taking like a partial derivative in Reverse so we're taking the we're integrating here with respect to X so we can treat Y just like a constant like it's like the number five or something like that so it really doesn't change the integral so what's the antiderivative of XY squared well the antiderivative of XY squared I want to make sure I'm color consistent well the antiderivative of X is X to the one-half sorry x squared over two x squared over 2 and then Y squared is just a constant right and then we don't have to worry about plus C since this is a definite integral and we're going to evaluate that at two and zero and then we still have the outside integral with respect to Y so once we figure that out we're going to integrate it from zero to 1 with respect to dy now what does this evaluate we take we put a two in here if you put a two in there you get two squared over two two squared over two well that's just 4 over 2 so it's 2 y squared - y squared - is 0 squared over 2 times y squared well that's just going to be 0 so it's minus 0 I won't write that down because hopefully that's a little bit of second nature to you we just evaluated this the two endpoints and I'm short for space so this evaluated a 2y squared and now we evaluate the outside integral 0 1 dy this is an important thing to realize when we evaluated this inside integral remember what we were doing we were trying to figure out for given Y what the area of this surface was well not this surface the area under the the surface on this kind of for given Y right for given Y that surface kind of turns into a curve and we've tried to figure out the area under that curve in the traditional sense right so this is going to be a funk this ended up being a function of Y and that makes sense because depending on which y we pick we're going to get a different air here because because obviously depending on which why we pick the area kind of a wall drop straight down that area is going to change so we've got a function of why when we evaluated this and now we just have integrate with respect to Y and this is just plain old vanilla definite integration what's the antiderivative of 2y squared well that equals 2 times y to the third over 3 or 2/3 Y to the third we're evaluate that at 1 and 0 which is equal to let's see y1 to the third times 2/3 s 2/3 minus 0 two-thirds times two-thirds well that's just 0 so it equals 2/3 2/3 if these were if our units were meters it would be 2/3 meters cubed or cubic meters but there you go that's how you evaluate a double integral there really isn't a new skill here you just have to make sure to keep track of the variables treat them constant when they need to be treated constant and then treat them as a variable of integration when it's appropriate anyway I will see you in the next video