Calculus 3 Lecture 14.1: INTRODUCTION to Double Integrals (Background Info)

so we have a brand new world I'm gonna give you an introduction on these things called now it's not a thing you're joking Leonard it's not no yeah double integrals sake ee ya see devil we can do double integrals now what in the world do they mean and how are we going to do them that's what this lessons about it should be a quick lesson alright remember where I spent a too long time on it because I want to get to the actual practice of doing double integrals but I do want to want you to notice the connection to calc one we're going to walk through what how come one is very quickly maybe five minutes then we're going to build double integrals from the calculon idea does that make sense so calc one the second part of the calculus question calc one first part was find the slope of the tangent line to cover the point we nailed out with derivatives we've done that a freaking time in this class the second part of that two-part question is how do you find the area under a curve and how we did it in calc ones like this we took a finite interval from A to B under a continuous curve and what we did was we cut we cut that interval into a whole bunch of little segments and of them so we took uh we took a 2b and we made a whole bunch of rectangles from A to B so basically we took a to B we divide it into an D flat equal rectangles that that was do get as a Surya belt - it'll it's called Ramon sums you get there's a second so okay so B minus a divided by n would give if I take this interval and I divided by the none of equal partitions in there I'm going to get the width for each partition this is the width of each rectangle as I clear for you that's how the math works here width of each of those they're all the same all the same way okay fantastic let's find one point for each of those sub intervals so at each point on here on the x-axis I'm going to have one x value we call that X sub K dot what that means is that for each one of these intervals one two three four five whatever I'm going to find that the X from each one of those so the X and the first interval of X and second X sub K X for each interval and the dot means it can be anywhere anywhere they're left right center somewhere between the typically particular center but it doesn't matter the dot means it's an arbitrary choosing at some point on each little value so cut it into a whole bunch of partitions how many n of them a lot of them a lot of them what's the width of each one well it's an interval divided by the end that gives you the width for each segment okay that's a width of a wrecked em you guys get it well to find area we need to weight that we need a height we're going to find a height let's pick a random point in there for each interval each interval pick a random point XK dot now find a height at each of those points okay fine well what for the history of the world gives us Heights how do you find heights of points what do you do look at it play into the function so if I have hold it to different intervals at a point for each of those on each of those little sub intervals and I want to find the height at each of those points I'm going to find the height at each of those points does that make sense now some of you guys are learning calc 1 for the first time because this is never shown to you I feel like what now I want to find the area for each of those rectangles area is width times height what's the width for each of those sub intervals that's what we call it they're all equal so if area is width times height I now have the width for each one they're all the same I now have the height for each of those rectangles does that make sense to you if you need the rundown again here it is each of these little sub intervals is Delta X Y every height can be found this way pick an X sub K X sub 1 2 3 4 5 see xo7 ducked your X sub 8 x of 16 pick up pick a random point in there X K done find the height basically just plug it in and then use that height as the height for the rectangle okay so I have width times height here's the height of each rectangle word should we're picking that to be the rectangle times the width that's the area of X each rectangle so hands feel okay with without that that that's it now add the moment how many are there in of it there's Hannibal and different segments so we found an X and different heights add them all up from K goes from the first one to the end point wherever you saw out of what will add up every single height times width every single area now here's the perfect thing about this because we're adding the same width over and over and over and over again you can factor it out basically this works this way add all the Heights you've factored out the Delta X vector of the width and all the Heights figure out that some x the width and that's effectively the same thing as adding all the areas to eventually okay we element now this is for sure different aesthetic in approximation you see it depends says how good is this approximation well you can see it's an approximation think about it if I divide this into a finite finite your keyword a finite number of say sub intervals finite number and I choose a random point in there for every sub interval and I find the height at every random point on every sub interval and I find the area by multiplying the width times the height getting area is there going to be error yes because this rectangle doesn't model perfectly the curve it doesn't it doesn't work that way so how do I make it better if this is if this is an approximation because I have a finite number of ends how do I make it better I need an infinite number of ends because that make sense how do I let this be an infinite number of ends kind of think backwards here I need to let the width go to really really if I let the width go really little there's an infinite number of pencil does that make sense now words you to put let n go to infinity let the number of rectangles go to infinity that packs in a whole bunch of rectangles it makes the width of these rectangles really really close to what number see really really close to zero so if I take a limit and say I want the rectangle the number of rectangles the number of subintervals are go to affinity and then I I do that then notice that K doesn't go from 1 to find a number K adds from every rectangle to the infinite number of rectangles Ament in that net space and what we get is no longer an approximation at all this is exactly so long story made shortly that took about five five six minutes hopefully here's a long story make sure calc one does this calc one says cut this finite area into a whole bunch of rectangles how many well ultimately infinity all right how to cut it up into equal equal lengths well then if I take B divided by the number of equal lengths I get my width Delta X find a point h1 cool random point that's the dot find the height of each of those rectangles well just plug in X K dot and use that as the height because that makes sense then we go okay take width times I get area add up all the areas no problem we can do that now that's an approximation to have a finite number of let it go to infinity now there's not a finite number now what happens is there's infinite rectangles in there and because you chose your point randomly doesn't matter where it is when you take that rectangle and you start squishing it that random point starts getting squished really tight right really really really close so that doesn't matter where you need it this is going to be so close together as a matter nope then we get like almost these are so close to zero but they have a little bit of bitty teeny teeny teeny teeny tiny width because if you don't have with you don't have area so they have a little teeny bit but doesn't matter where that choice was XK dot this is that this right here is the definition of what an integral was an integral is a limit of a sum it's why we get an S that's an integral it's an S it's a sum an infinite sum is an integral of what a function with X as the variable chosen random but there's a matters anywhere along the x-axis any time we let a DX approach to us or I Delta X approach 0 we get a DX out and that's where the stuff is coming from and infinite sum of a with respect to X as the width between those two numbers that's magic that's all of Calphalon a nutshell for various and they go from B to B that right there is the definition and how integrals are even constructed to make sense to you let's got one now let's go to calc three all right what's the idea now well one variable was area under a curve it's two variables going to give you can't be area one variable area two variables extends the dimensions if one variable gives you curves two variables gives you services if one variable gives you area two variables gives you volume right now the idea is find the volume find the volume underneath the surface as bound by a region find the volume under a surface that's bound by a region you have to extend everything one dimension does that make sense that's cool that's we're about to do I'm gonna build it for you it's not hard but you need to see at least once to see where it comes from okay so so here's the idea we're not finding height and x width anymore because then I'm going to give us volume we're finding height and multiplied by area surface area of every little region down there that's what we're going to be doing now so I'll build it for you let's take the region under some surface so here's my surface I cut a little region and I'm going to I'm going to set this down here and right now for right now I'm just doing with rectangular regions to make our lives easy I'll show you how to do with non-rectangular again right now we're just dealing with some rectangular regions can you see that on the XY plane I'm going to have a rectangular region though yes don't yet that's what this is so X goes from A to B X goes from A to B Y goes from C to D Y goes from C to D are you guys with me now instead of instead of segmenting this just along the X I now have to segment this just along the Y so I'm going to I'm just going to choose this to cut this twice and cut this once it doesn't matter the more cuttings obviously the more accurate but just to give you a picture of this I'm going to cut this into two segments here and one segment here the key is the key is just like I wanted equal subintervals here I need equal rectangles here do you guys see that the extension of the dimension just like calc 1/2 cup 3 does instead of just adding length now that length times width so just having a distance here and now I have an area instead of just did an area now I get a volume so I'm going to cut this here and here so we're going to take that rectangular region and we're going to cut it but the key is I need the rectangles to be equal size equal equal double evil the number of partitions that you cut the x-axis in is n the number of partitions you cut the y-axis in is n so in this so M represents the number of partitions along the X and number of positions along the Y you guys okay with the terminology there notice all the rectangles are equal that's what we want here number of X partitions an abbreviation why number of us are n number of Y partitions everyone in the room right now please how many what is our n what's our value of n here is it three or Z 2 it goes along the x-axis how many out with those are n in this case everybody class right now because only the x-axis ends along the X what's our value of M stolen exits it's so helmet so here's our interval from A to B how many time how many partitions do I have from here to here looking on how many out let's count it here's one here's two here's three and it's three chemistry we have three partitions going along the X now let's do the Y I'm going to partition so we have going on why is it perfect so your n is in it and the mmm-ma the an is-3 the end is yeah that's right that's right so hence real good fella that's the idea it now now just like here we picked a point I really point though an x value can pick a point along the x-axis the next value from that tensions pick an x value for every single subinterval here we're going to pick a point for every sub rectangle you guys get to pick a point on every one of these how many points are we going to pick for this specific example how many rectangles do we have you're always going to have M times n rectangles do you see it get the X and M because the wide end M times n gives you the number n times n my M&Ms are off okay on like M times n gives you number of rectangles you're going to pick a point for every one of those this is going to look crazy to you but watch carefully instead of X K well now we need to be able to do this we need to go this rectangle then this one then this one then this one and this one to this one if we had a bigger grid we go boom boom boom boom boom boom boom boom and we count up all of them so we need X sub IJ y sub I J dot as they got to be able to be arbitrary here's what this means because it looks really weird but here's what this means you would do X 1 1 as R X x coordinate of the first rectangle along the X and along the Y and Y of the first rectangle on the x and y so you count X in the first rectangle and line the first rectangle but then you go X in the first in the first rectangle on the X and the second rectangle on the Y so this would be rectangle 1 1 this will be rectangle 1 2 this will be rectangle 1 3 rectangle 1 4 if we had it this would be rectangle 2 1 2 2 2 3 2 2 3 1 3 2 3 3 2 get it and for every rectangle 1 1 1 1 1 2 or 2 you pick a point so somewhere in there some X Y in the first rectangle second rectangle 1 2/3 yes them pointer clicking the right finger on the boundary or yes typically we choose corner points for the center point but it can be anywhere it the dots that means were arbitrary like this and you can imagine if we let all these rectangles get really really really little doesn't matter we're a bigger point makes sense doesn't matter because we're going to do that so n times n rectangles very good for each for each rectangle we find an arbitrary point pick a point no problem what you suppose we're going to do now well here we picked a point and then what we do come on what we do at the point what we do with the value I should say what yeah then to get a height okay so let's plug it in to get a height find a height well that would be functions always give Heights I know the notations crazy because you're not used to dealing with like double indices but I do want you to be able to follow it how we're going on rectangle one one rectangle one two rectangle one rectangle two one rectangle two and every time we're in there that's all the Incans guys in a given rectangle you're picking up on that's all I'm saying and you've got to pick a point for all six of these rectangles are all infinity of the future rectangles okay now we find a height all right so it's so easy we get that we typically is was asked where we better point we typically choose to pick one of the corner points or the very direct dead center to make our math easier it's like oh I want you to pick that point right there how you going to describe that that's really hard here it's pretty easy to take this and divide by 2 take this divided by 2 for the center here it's really easy if you want to make a quarter point so we typically do that now finally are we looking for area or ball for me is the surface area of every rectangle and they're all the same times the height of each of the function at each arbitrary point so volume equals surface area times height let's work with that let's work here here's the math if X is an element of on the interval a B and Y is on CD let's start building this in there somewhere so we're taking a little foamy alone cat I don't wanna lose you here you've seen too much okay you seem to me you gotta learn it all now this was just along the x-axis right I took a to be I cut it into N in this case x @ n with it and intervals the width of each one was B minus a divided by N and a gives Delta X correct we're now cutting x + 2 n just a different changeable it's just a letter change we're cutting X the interval of X into M sub intervals so that's a still goal for me to be we're still cutting it different letter but is still cutting it this is going to get look carefully look carefully because it usually where all the money comes back this is going to give us the width of each rectangle along the x-axis does that make sense that is the width of each the width just like this width of each rectangle a nice axis but now we're also cutting live they might see into how many how many cuttings here for why how many oh this is going to give us the if this is the width then this is the height of each rectangle longer wife still with me made sure so so again the idea drop it down rectangular region cut it in a whole bunch of rectangles how we going to do it wealthy colored rectangles then there's going to be a certain number segments along necks certain on flip segments along the line cool the width of each one here is Delta X this is your weight the height of each one notice they're all the same the height of each one is double one we are almost done trust me almost done hey if you're rid of a rectangle as Delta X and your height of the rectangle is Delta Y can you find the area of each rectangle boom how true or false the area of all those rectangles are identical just like the width here was already the area of each rectangle built x times Delta Y Rama's done the cool part if volume is surface area area just a carry if volume is area times height do we now have the area right there do we now have the height above each rectangle at a arbitrary point yes therefore the volume the book please listen the volume so that this is under here right we've cut it like this the volume of each one of those rectangular prisms we pick a point somewhere in here we find the height of that area times height gives us a volume the volume of each of those little column cubes if you want the rectangular prisms with a weird top the volume of each of those rectangular prisms is hype and an arbitrary point for each rectangle get it times the surface area of each rectangle do you see that's giving you volume look at the board look like or well so width times height of the rectangle times the other height length times width times height that's going true/false this is exact that's false this is the volume please let's carefully this is the volume of each rectangle just like this right here is the area of each rectangle how many rectangles do I have a lot of them what are we going to do with the area of each rectangle we're going to add it up we're going to do with the air the volume of each rectangle what are we going to do so if I have the volume of each rectangle then the volume of the entire area under the surface is going to be let's add them up but way to say animal love but here's the whole magic here's where a double integral comes from you're not even listening reversing its mr. cool part it's so cool part is the coolest part gosh darn it Oh second mister speaking in squeaky voice you're a monster I practice that on the weekends you watch kiddie movies I practice all the weird voices Oh her they're really naughty look if you're adding across two different dimensions you need two different sums we need to do this we'd be able to do have I want to run yeah I'll do it this way we need to be able to let our eyes start at 1 and go to M for X and let our Jays start at 1 and go to N or why I want to talk about this I want you to understand this I don't want you this don't skip this ok don't miss don't miss this once you have 7 what's right 7 it says you're going to start at the very first rectangle but for every time you have the I equals once you understand that I would go 1 2 3 4 5 10 J we go 1 2 3 like a grid you can see that for every time I get one of these I have to run through the whole gamut of those so for I equals 1 I go hello I equals 1 J 1 I 1 J 1 I 1 J 2 pi 1 J 3 and up all that that row column had don't call then I have to let this go to 2 but when I let me go to 2 now - sir - 1 - 2 - 3 - that L 3 3 1 3 and that's how we do it we add up a double sum that's what we're doing we're going up the grid of rectangles never switch make up the grand again that's what that says and not forgive none now we'll ask is this exact that's an approximation because I have a finite number of oh my gosh for all the money in the world because I don't have how do I make it and only make it better if that's finite I wanted the Wits to go well in other words I'm sorry I want a number of rectangles to go to infinity I want the number of rectangles to go to anything so this is an approximation because it's finite I'm gonna do over here it's too too important for me to squash down there that's going to squash too I'll have to race it but I want to make sure you saw the connection from here to here you guys see the direct connection see if you pay attention some you guys have your eyes closed pay attention oh goodness gracious exact or not approximation how do I make an exact I let the number of rectangles go to infinity which means I let this go to infinity the number segments along X go to infinity and number signals along Y go infinity means that the arbitrary tubba to visit arbitrary point doesn't matter because a rectangular V is so close together but matter where you pick it and the way I do that I take a limit however the limit does two things the limit lets both m and n m and n both go to infinity cool right now I'm applause right here before I give you the final punchline which is which is it that we do like two examples we would call this thing good do you see where everything's coming from because honestly a lot of people get to this level and they've never even seen this before which shames me I do it in calc 1 it's there but some people don't there's go always how you do integrals that's crazy okay you need to know where that comes from is what I gave to you do you see where that comes from let's let's let's look at it what is that area this is not area this this particular one with the lung max what's that this is the area of each rectangle this what's this without them without that without that that's a arbitrary point on each rectangle with me the path gives us a height so this is area of each rectangle times height that's the volume of each little prism as I add them all up across the grid and let the number of them go to infinity the arbitrary point doesn't matter where if you think it's going to end up being squished in the very middle of that American stood so small it sits in there somewhere and then I add them all up after taking that limit that gives me not an approximate volume anymore that gives me an exact volume that's calculus that is what it is it's that idea right there and we make it easier we do we do some definition so you know what every time you take a limit of a sum talked about it right here the limit of a sum gives you what so the limit of two sums gives you two this is two integrals a function of an arbitrary point gives us a function with two variables every time we have a delta X going to and notice if an go to infinity the width between them go to zero every time we get a Delta X Delta Y going to zero we have a DX and a dy that's what that represents it says the distance between them is going to zero you're adding up this this this is exactly what this says it says that over some region over some region I am going to add up add up across the grid a whole bunch of rectangles where this is the height of each rectangle and that is an area that's approaching zero of each rectangle double integrals both that's what double integrals means so don't lose it don't just start doing calculus you like I'm just doing double integrals what do they mean what are you doing every single time you do don't living finding the volume under a s bound by whatever region that says so far we've dealt with rectangular regions I'll show you how to do with non-rectangular at the very end we're not done yet but that's it I'm also going to try to kind of prove foo be nice theorem right now but you don't even know that means but you will shortly in 14.2 there's a question does area matter in which order you multiply so if I have this for this will I get the same result the beanies that four rectangular regions approved it's basic it's multiplication bold-faced community proofs medians their mother yes since reactor Vera horizontal integrate that or we have the integral with respect to Y and then the road will just respect to us that's a question for the next section yes remember partial derivatives how would you do the opposite of a parcel groovy one than the other piece of cake just sort of oh that is the same place and gentlemen do you understand we're double integrals come from this point now I would be remiss if I didn't give you at least like two examples of how to actually do this with the Romans some sort of idea so we're going to do that that that's good it's going to be fast Leonard's version of s I'm going to show you well you'll see so let's do one example because I want you at least see it where it comes from and see the working of it are you guys with me on this too bad you're gonna do your new daily you devil please all right because I want to this is the good stuff this is the this is the bright point and you're mad today right here when we get to practice I want to find an approximation for the volume of the following function on a given rectangular region it looks pretty wordy trust me this this stuff is really easy because you are literally just plugging in numbers and adding things together it's really easy however just like before in order to do anything with any you're listening now but you should be put this in the back of your head anytime you do anything with anything involving a region just like in thirteen point eight anything that means don't wanna lose that means right here draw the region every single time draw the region so we're going to start that right now this let's draw the region notice how x and y create for you this knifes rectangular region X goes from 0 to 1 Y goes from 0 to 2 and we have this semi rectangular region it also says this I want to cut this and this is why you need to know what M and M means I want to cut the x-axis into two segments equal segments and the y-axis into two segments equal segments we're now creating we're now creating that's like a book of it here's a picture a bird's eye picture of the XY plane the surface is over that somewhere you can tell it's a service over the XY plane now instead we'll look at that service it's going to make a cutting on the surface find the volume beneath it how do we do it let's cut that's that region underneath the surface let's cut that in some rectangles how many two each so two cuttings each hey hello this look at the word what's that if I cut this in half what's that what's this and I kept this I can do just cut it no problem I get cut into tenths if I want to but I'm not that that statistic so you'll figure one fractions enough for this now the rest of it it's literally just using some formulas literally so let's find the width of each rectangle you can tell it's going to be one half the width is one half the height is one you see it if you didn't know how to look at the picture and do that you could do just from Delta X you go well well Delta X says that I'm going to take my eggs bounds here and divide by the number of cuttings for X that's two you have to get 1/2 as our picture shows us it's got to be Delta Y says let's take our bounds for y let's cut them by the number of cuttings for y let's get it to a 2 and that gives you 1 let's furthermore find the area for every one of those equal rectangles it's easy you delecate with that we have a special special thing for this we typically don't call it just a we call it Delta a because that area is if we let it go to a limit that area is going to shrink to zero also if Delta X and Delta Y are both shrinking to approach zero Delta a is going to shrink to approach zero does that make sense to you I know it's new notation but it doesn't doesn't really affect us that much now how many rectangles is this create for you we're going to find the area using lower right corner points lower-right point man i just rally up at certain points lower right corner points so on your grid why you draw the Legion on your grid let's look at this can you tell me where the lower right corner point is the coordinates we're there there there and there if there's four rectangles you may only put down four points points equals rectangle number points equals member rectangle so that makes sense we typically choose corner points one of these or center points that's the hardest and not super hard but it's the hardest to do and then well follow the formula but the formula says is hey look I want you to plug in those points which plan I've added up first because all the errors would say multiplied by the area last so our volume I'm going to cup of equals because it is an approximation our volume is going to look like this let's find F list out let's go slowly but list out my corner points the first one has been second one third one and fourth one can you the set the cord it's for me please one half okay and then the next one I'm going to plug in is one-half zero this coordinate is 1/2 1 okay let's practice this one everybody what's this coordinate and then lastly this coordinate please watch so I don't lose you please watch what we've done right here this is a point arbitrary map and picked it doesn't matter we picked a point for each rectangle you see it for rectangle square ones we plugged in the points to the function what to do when you thought about going to function what to do hey we just found the height at each point how we find volume is height times area only I'm not going to do it every single time because if I multiplied every single one of these by the same area I could easily factor it out someone add up the heights first and then multiplied by what what area if you want what area specifically what area specifically will it marriage because that's that's my area for each rectangle does that make sense now I've done the work ahead of time so we don't burn up our calculator batteries and stuff we can plug all this stuff in if you want to check your work later 25 pounds to plug it in later if you want you all let Dennis took this pull you to the function and figured out the results that it's just plugging numbers and all a short time doing that okay what's it represent what is that what's the whole idea of a double integral what are we doing volume under that surface or the I am going to but only over that rectangular region my sense is it exact that begs the question if I change my points will it change my body absolutely because if I change my points I'm plugging in different values to that function it will change as a matter of fact if you want to try this on your own if you plug in centered points here here here here that would be 1/4 1/2 3/4 1/2 1/4 one and a half three quarters one and a half G see it plug in those values the volume is approximated by 49 a report you want to check your work on that one try it try it later on your own time do now try later see if you can plug in separate points and get 49 over 4 it's off a little bit it'd be off a little bit thing that's it's not the same it's close that's not the same this is 50 over 4 Center points gives you 49 for slightly different try it am i making sense kind of goat if I changed that around you gave you four could you still do it I change that around give you six could you still do it I change this around and went from one to five could still do it picture looks a little different you have a different number of rectangles here you have four rectangles two times two if I give you four and sixty-eight n rectangles that's ten points to plug in what they did remains exactly the same I don't want to do another example because they're meant exactly the same I choose you tell you to choose left lower left corner points or upper left corner points or upper right or center points do you feel like you could do the work off because because actually I just wanted to get down to here and do one to show it I'm going to show you one more thing to answer any questions after that okay so this also bases you guys probably have the question your head our all regions rectangular yes okay can you draw me a region that's not rectangular yeah I can rectangular yeah vixen we learned a lot in geometry oh wait we don't teach geometry here oh alright ladies and gentlemen I'd like to introduce to you a closed bound figure which is not a polygon nor is it even a rectangle this happens to view just a simple closed curve a region on the XY plane with me not a rectangle however verify this just logically if it's closed which is it can be bound by a rectangle rectangle round I tried my best to do a big rectangle keep it bound by a rectangle we make it we make it as close as possible we try to contain it on the edges I just got a little sloppy gears to show you something here and then we go okay let's take the rectangle region that's that has bound this non rectangular region and let's cut it up into some rectangles equal sized rectangles and now let's find a point on each of those rectangles on the two center points is for funsies to show you some stuff have to find a goalies here's Jenny here's ready even if it's not rectangular we can put a rectangle around it correct here's the magic of it could you still given this rectangle on this grid what you have could you still plug in these points from that rectangle I'm going to have coordinates for those points yeah okay still find the height of the function above that point so here here's the whole deal if even though this goes outside of some right angles if the point you're evaluating is in the region or on the boundary you include it if it's not that you don't so would we include this region or this rectangles volume in the sum of this one how this one this one this one is the dot in the region this one this one this one this one yes yes yes yes yes yes on the boundary counts yes yes yes yes yes yes yes yes yes yes no no no yeah that's how we get around the fact that we don't have rectangular regions we make them bounded by a rectangle and then if you can imagine this we're going to infinite number rectangles in there right if the number of rectangles means those boundaries are all going to be Center points every point it's giving some center point of a rectangle they're all going to be including everything and then if there's an infinite number of them we add em all up that's a volume of a service over a non rectangular region and it's exactly the same it's the same you can see it's the same thing it's taken all the rectangular system rectangles guys so right now you should still add them up across big ole grid it's just if the points you choose are in there you don't include them well that doesn't matter because if I take infinite number of them all the points are going to be included that that's that's the point also I'll prove Ruby nice theorem for general regions ready for it does it matter if I add them up this way if I multiply does it matter switch around for non rectangle there proof abuse theorem twofold what is this stuff cool or what that's cool oh yeah and we're done so that's good

Channel: Professor Leonard

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Published: Mon Apr 11 2016