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# Triple integrals 2

## Video transcript

in the last video we had this rectangle and we use a triple integral to figure out its volume and I know you and you were probably thinking well I could have just used my basic geometry to figure out to multiply the height times the width times the depth and that's true because this was a constant valued function and then even once we evaluated when once we integrated with respect to Z we ended up with a double integral which is exactly what you would have done within the last several videos when we just learned the volume under a surface but then we added a twist at the end of the video we said fine everything you could have figured out the volume within this rectangular domain I guess very straight forward using things you already knew but what if our goal was not to figure out the volume our goal was to figure out the mass of this volume and even more the mass the the material that we're taking the volume of whether it's you know volume of gas or volume of some solid that it's density is not constant so now the mass becomes kind of I don't know interesting to calculate and so what we defined we defined a made enca T function and Rho this P looking thing with a curvy bottom that was that gives us the density at any given point and at the end of the last video we said well what is mass mass is just density times volume or you could always you could view another way density is the same thing as mass divided by volume so the mass around a very very small point and we call that D mass the differential of the mass is going to equal the density at that point or the rough density at exactly that point times the volume differential around that point right times the volume of this little small cube and then we as we saw it on the last video if you're using rectangular coordinates this volume differential could just be the X distance times the Y distance times the Z's distance so what if we wanted the density was that our density function is defined to be XY and Z and we wanted to figure out the mass of this of this volume and let's say that our XY and z coordinates their values let's say they're in meters and let's say this density is in kilograms per meter cubed so our answer was going to be in kilograms if that case and those are kind of the traditional SI units so let's figure out the mass of this variably dense volume so all we do is we have the same integral up here so the mass it for the the differential of mass is going to be this value so let's write that down so it's let me it is X I want to make sure I don't want a space X Y Z times and I'm going to integrate with respect to DX DZ first but you could do that you could actually switch the order maybe we'll do that in the next video we'll do DZ first then we'll do dy then we'll do DX and then we just define so this is this is once again this is just the mass at any you know small differential of volume and if we integrate with Z first we said Z goes from what the Vaught the boundaries on Z were 0 to 2 0 to 2 the boundaries on Y where 0 to 4 and the boundary is on X X went from 0 to 3 0 to 3 and how do we evaluate this well what is the antiderivative of we're integrating with respect to Z first so what's the antiderivative of X Y Z with respect to Z well it's let's see this is just a constant so it'll be X Y Z squared over 2 right yeah that's right and then we'll evaluate that from 2 to 0 and so you get I know I'm going to run out of space so you're going to get 2 squared which is 4 divided by 2 which is 2 so it's 2 XY 2 XY minus 0 so that's it so the the evaluate when you evaluate just this first interval to get 2xy and now you have the other two integrals left so I didn't write the other two integrals down maybe all right now so then you're left with two integrals you're left with dy and DX and Y goes from 0 to 4 and goes from zero to three I'm definitely going to run out of space and now you take the antiderivative of this with respect to Y so what's the antiderivative this with respect to Y let me erase some stuff just so I don't get too messy I would I given the very good suggestion of making it scroll but in this unfortunate it make it scroll enough this time so I can delete this stuff I think oops I deleted some of that but I you know what I deleted okay so let's take the antiderivative with respect to while I'll start it up here where have space okay so the antiderivative 2xy with respect to Y is it's y squared over two twos cancel out so you get XY squared XY squared and Y goes from zero to four and then we still have the outer integral to do X goes from 0 to 3 DX and when Y is equal to 4 you get 16x 16x and then when y is zero the whole thing is zero so you have 16x integrate it from zero to three DX and that is equal to what 8x squared 8x squared and if evaluated from zero to three when it's 3 8 times 9 is 72 and zero times eight is zero so the mass of our figure the volume we figured out last time was 24 meters cubed I erased it but if you watch the last video that's what we learned but its mass is 72 kilograms and we did that by integrating this this three variable density function this function of three variables or you can in three dimensions you can view it as a scalar field right at any given point there is a value but not really direction and that value is a density but we integrated this scalar field in this volume so that's kind of the new skill we learned with the triple integral and in the next video I'll show you how to set up a more complicated triple integrals but the real difficulty with triple integrals is and then you I think you'll see that your calculus teacher will often do this when you're doing triple integrals unless you have a very easy figure like this the evaluation if you actually wanted to analytically evaluate a triple integral that has more complicated boundaries or more complicated you know for for example a density function the integral gets very hairy very fast and it's it's often very difficult or very time-consuming to evaluate it analytically just using your you know traditional calculus skill so you'll you'll see that a lot of calculus exams when they start doing the triple integral they just want you to set it up and they take your if they take your word for it that you know you've done so many integrals so far that you could take the antiderivative and sometimes if they really want to give you something more difficult than to say well change switch the order you know this is the integral when we're doing with respect to Z then Y then X we want you to rewrite this integral when you switch the order and we will do that in the next video see you soon