Changing the order of integration of a triple integral

okay this video I'll show you guys how to change water food equations of this triple integral right here and we have busy dy/dx and we're trying to get to DX dy DZ so of course were to figure out these right here and in fact we have four other ways to organize the dy DX and DZ right that if ratios and the reason that we have a total of six it's because for three different shots we can just do three factorial that's three times two times one that's six so there will be a total of six ways to organize that but anyway in this video just show you guys how to do it with respect to this water in the meantime please pause the video and try this first okay okay as you all know the hardest part of doing all this is that we have to be able to visualize the solid that we're talking about and the reason sisters it's because we have a triple integral and that's a three dimensional thing right okay I'll show you guys how to do it hopefully the equations they are not that bad and I'll show you guys how to do all this by hand okay including a joint anyway here we go I will stop with my past the x-axis right here and positive y-axis and then past the z-axis and this is how we were told us through the picture okay first of all we have D Z so that means we have Z goes from 0 to Z equals to 1 minus y so we don't have any X at the moment so we can focus down the Y is the play ok and we see that when we have T is equal to 0 this is the z axis these secrecy roots right here but this equation in the space it's actually the whole plane right here right and in fact that's the XY plane you can imagine it's just like the room and this is lighter for of the room I'm just going to kind of shade this in to indicate we have pretty much oldest okay so just to this kind of things this right here where we present a whole play whole XY play and the reason I didn't go to the other side right here it's because I noticed there's no negative Y value so I'll focus on this part here now focus only why it's equal to 1 - why let's just like a linear equation on the zy plane right and to graph that ok we can say when y is equal to 0 Z will be 1 minus 0 which is 1 Y is equal to 0 it's right here and I say Z is equal to once right here so we have a first point and then when y is equal to 1 1 minus 1 0 for Z so let's say the 1 is right here for y and z is equal to 0 it's right here so that's the second point connect the dots this is just like a equation of the line I will just connect the dots right here except for the fact that if you're talking about X Y Z like the whole space it's not just this like this line actually popped out right it's a whole plane right it's like and then you also go into the wall and to indicate that I will just do this for you guys so pretty much you can just draw a bunch of this parallel line to make it three-dimensional looking okay so pretty much just do all this so hopefully so far so good what we have now it's the whole blue of a room and then like another like diagonal cut like that all right well done with that now moving on to the dy and we have this right here why it's equal to x squared up to what's equal to 1 okay we are talking about the XY plane now and you don't just tell your head a little bit to graph y is equal to X square as we all know that's a parabola but when you graph this do it carefully so I will just kind of once again turn my head to get a little bit like this because this is the y-axis this is the x-axis so the parabola will look like this ok yeah something that so we start with this parabola and this is just like the base okay the truth is you pretty much make the vertical cut okay like this so what you can do is you can see I said once you have the base on a plate you can just copy and paste many of this so you can kind of just make this into a three-dimensional looking picture okay it's kind of hard to draw but hopefully get scared my idea by anyway it's going to be cut by this diagonal line the second anyway here y is equal to what that's nice because Y is equal to one on the XY plane it's just this line right here okay so for the base who have this parabola this portion of it and the last part best parts easiest part the X that means X goes from negative one to one and that's exactly the intersection right here and right here this line was y equal to one and the curve was y equal to x squared right so you know this right here is exactly if you extend the x axis back to negative then this is X equal to negative one and then when you do it right here this is when X is equal to one okay so it's pretty much hold this protein and then like why exactly am i talking about image to another legitimate picture so this is how it looks like first of all we have this C value goes from zero to one so that's like the height of the solid right so I would just draw it like this and the base is this portion of the parabola so I would just do like this right here and then in the back it looks like this right and then remember this is not like the pattern and the base the pattern at the top of the same it's I've been cut it like slanted right because of the equation C is equal to one minus y so to connect the dots you go from here and you don't draw a straight line you draw like a prop like this way okay and then you do like this right here and we have that line right there so it's pretty much like this right okay and we can just color this a little bit so hopefully you guys see this is more three-dimensional looking so it's this kind of stalled it anyway here is the work now we are focusing on X writes from X to Y so we are looking at the base look at X first as we all know right here y is equal to x squared we can just square root both sides so that means X is equal to plus minus square root of Y and when you are going from X to Y you are looking for this phone you have to write in the back right here this equation this portion of the parabola has the equation X is equal to negative square root of Y right and then when you look at the front portion this is the positive version this is X equal to square root of Y and of course we're just label this right here X goes from negative square root of Y up to X equal to square root of Y and that's it for the X values all right now from Y to Z well well let's see of course why start testing off right here so that's why could a 0 right here and if you look here from Y to Z you're looking at mainly this line right and this line earlier has the equation C it's equal to 1 minus y right and then from this equation we can just move the Y to the other side so we have y equal to we have to want and put Z to the other side so becomes minus D like this so that's pretty much from Y is equal to 0 to Y it's equal to 1 minus Z so that will be right here y equal to 1 minus Z and lastly we are looking at disease right so that's the Z values and from either picture you know this right here was when Z Z could roll and the height as I said it's one so this is pretty much it C goes from zero to one that's pretty much it okay so I'll show you guys this and hopefully it gets all like this and the challenge is that can you figure out the other ways for example maybe I will just leave this to you guys how would you figure out in Turku in Turku integral if in this case okay it's the same situation with the same general function but I want to have let's say the Y first and then DX and then DZ okay but as I said these are just view ways and we have stream other ways other than this it's a good exercise for you guys to think about it but I won't lose to you guys and you guys can come and download all your answers and everybody will participate how is so excellent right anyway that's it [Music]

Channel: blackpenredpen

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Keywords: blackpenredpen, math for fun, bprpLIVE, Change the order of a triple integral, animation of a triple integral, re litterate a triple integral, calculus 3 integral, Changing the order of integration of a triple integral

Id: -pOV7n8qlbY

Length: 9min 40sec (580 seconds)

Published: Sun Apr 08 2018