Definition of the Integral (review)

Start   we want to remind you that these things we're going to use them from time to time if you dont have one and you have a smartphone or a laptop or something like that you can use that instead if i show you, how many people do not have a clicker?
or have a web enabled device of some sort with them ok do i need to show you or can i just tell you?
i'll just tell you umm
0:32here goes the class page which you should be able to get to through blackboard or go to this webpage youre typing all this crap and you can put clicker after it or theres a link on this one it says something like clicking without a clicker or something like that
1:04okay so you can just open that page and then theres a link about something like that ok?
so in fact since you have your clickers here lets have our first clicker question
1:37so A is also 1 should be 41 we're going to try it you can still do it
2:02and theres people answering yay so the way the clickers are graded by the way you get 1 point for answering and 1 point for answering correctly this might be correct right because if you have so well its alright you should be able to figure out
2:32the difference between these 2 things for some of you this is the right answer for some of you this is the right answer oh yeah because im stupid thank you so for those of you who dont have one i screwed up thank you for reminding me uhhh i have to launch my web browser and start the question duh
3:00so for those of you who dont have clickers itll say theres no questions open because the network doesnt like me at the moment but it will so please be patient okay so for those of you that dont have clickers if you click the try again button you should be given a choice yeah?
the question doesnt come up, theres the question it just says theres a-if you dont have a clicker and youre using uh a smartphone or something
3:31you should have a bluish page and then it should say enter your answer and also enter your ID number okay did everybody manage to figure out how to answer that i have 97 responses for the clickers you just press the number and then you press 1 2 or 3 and you can press the little thing in the center that says send and it should have a little smiley face if it has a little smiley face it means i got your answer
4:02so for those of you who did the web poll it should say i took your answer ok everybodys answered right?
except him he hasnt thats okay dont worry you'll get a chance later okay so im stopping this question now umm and so i can see here im not going to put it up unless it matters um 99% of you answered #1
4:32um 1% of you answered what?
and those of you with phones and whatever i dont get to see your results until i go back to my office you should have all answered no if you answered yes well i guess yes could be correct it just means its not with you but anyway okay alright so i will do this not stupid questions-well a lot of stupid questions like this too as we go but this is just a warm up
5:02i wont necessarily ask clickers clicker questions every 5 minutes i may only ask 1 i may ask 0 i may ask 10 whatever the point of this is to just get you to interact with each other and stuff uhh yeah?
oh thank you umm ok so now we need to do some calculus-well there was one other administrative thing and i just forgot what it was
5:31okay so uhh web assign homework there is a homework assignment on webassign its up its been up for awhile it was it it was due on wednesday at 9 am because of the hurricane i added a couple of extra days it is now due far too early saturday morning this means do it before you sleep on friday night
6:00if you dont go to sleep on friday night well then you can do it saturday morning and if you do any questions that you answer uhh i dont know, early by the original due date ill give you extra credit so the grading online-i explained this last time so dont be confused the way the grading works is if you answer right the first time you get full credit
6:30if you answer right the second time you get 1/2 credit if you answer right the third time you get 1/3 credit and so on but if you answer right the second time but early then you get more than 1/2 credit because you get a bonus on your 1/2 credit you get 3/4 credit yeah?
no if its more than 2 days before the due date
7:02you get the credit you get the bonus so you dont get a bonus over doing it right now versus waiting until monday generally if you do it over the weekend because theyre usually due on wednesdays you get the bonus but if you wait until monday or tuesday you dont so the main point of the bonus is to get you to think about the problems do those you can do and then worry about those you have trouble with later
7:31i know there was another but i cant remember what it is you dont know what i covered umm alright by the way this is professor bonafant i dont know why shes here but she is she teaches the other lecture at 5:20 if for some reason you dont want to come to this lecture because im too disorganized or whatever umm okay so we need to do some math now i know it seems confusing but we'll do it anyway
8:10whats calculus?
does anybody know?
nobody can tell me? yeah?
well kind of not really so he says the study of rates of change thats not wrong dont think you'll get what im trying to get here so what calculus-anyone else have an idea?
8:30so youre righter than most no?
nobody knows what calculus is?
youve all had at least a semester of calculus and you dont know what it is?
he knows hes the only one thats been in class yeah?
okay, or a rock so. okay umm so what it is about the main idea the big idea in calculus
9:07the huge idea in calculus is that youre going to understand macroscopic stuff by knowing
9:31umm microscopic detail that doesnt tell you what calculus is but thats the big idea this is the big amazing breakthrough that Newton and ??? had Newton was supposedly sailing on a river and he wanted to understand how his boat was floating on the thames or something and he wanted to understand why his boat was moving and he was looking at the currents and realized that the currents were pushing him and if he just knew how all the currents fit together
10:02he would know where his boat was going to go and this is the big idea that makes calculus work and the way that we understand this microscopic detail in modern days is via the notion of a limit limit probably you know in many many high school classes and so on and actually in college classes
10:31a lot of students-this idea of a limit is just that stupid thing you have to do at the beginning of the year before you can get to doing the real stuff where you take derivatives by formula but for mathematicians this is the thing this is what calculus is if theres a limit there its probably calculus mathematicians tend to call this analysis but if theres a limit theres probably some calculus hiding somewhere umm
11:00so these macroscopic things that we want to understand are really functions so we have some function we have some function and we want to know stuff about it and we want to use this microscopic detail to learn things about it
11:31what you all studied in your first semester one of the things that you studied was the notion of the derivative the microscopic thing that we're looking at we take our function-let me do it over here and we blow it up and there it looks pretty straight its not quite straight so we blow it up a lot more and eventually it looks like a straight line
12:01so this is the notion of the derivative which you are all intimately familiar with if not youre in the wrong class go away now umm is this idea of blowing something up and zooming in until it looks straight what are we doing when were zooming in we're taking a limit?
we want to understand whats going on at this point and in order to understand that
12:30we zoom in zoom in zoom in and this process can be formalized by taking the limit as h goes to 0 f(x)+ h lets call this a instead of x over h and this gives us a thing that you know about but you probably forgot this formula
13:01of the derivative this is not what were talking about in this class but this is what we will use all the time and so this is one of the big ideas of calculus is the idea of a derivative whats another big idea in calculus?
integrate okay so integration the idea in integration the idea in integration we want to ask a slightly different question about the function
13:34not how this is looking in the small we maybe assume that we know about whats going on in the small we want to know its sort of the same function we want to ask the question what is the area here?
and we want to use the same idea
14:00um this idea and in order to understand the big question the macroscopic detail we want to focus on the little things and then use those little ideas to put it together to understand the big thing so in order to find the area here this is my graph y=f(x) and i want to find and im not going to shade it because we want-well i'll shade it
14:30we want to find the area there and again something that you all should know and if you dont know then you should be in math 126 instead of 132 is.. the way to find this area is to chop it up in tiny little pieces im going to use big pieces but lets just chop it up so i chop it up into pieces and instead of finding the area of the whole thing
15:00i just find the area of the little thing well.. if i look at this piece here it looks almost like something that i can take the area of lets chop it up some more so maybe i chop it up a little more and if i keep on chopping the pieces the pieces are.. very skinny
15:30and then they have some little bit of curve on top of them and i dont bother thinking about whether this curve is tilted down or tilted up or whatever i just think this looks i mean to you i can see that i drew the top tilted but i doubt any of you can maybe you guys in the front can so this is a rectangle just like in this case
16:01this bit thats curvy if i zoom in enough it looks pretty straight here if i chop it tiny enough it looks like a rectangle rectangles are really easy to find the area of you just take the height times the width and we know the area and so if we chop this thing up into tiny enough pieces all we have to do is add up the area of rectangles
16:33and that gives us the area of the big thing okay? so what is the area of this rectangle?
this should be a clicker question but im not going to write it nobody knows really? come on right, its the height times the width
17:04whats whats well we'll stick the widths for a minute whats the height well it depends on where i am but if this rectangle sits right here and theres some point here lets call it x star sitting in there then whats the height f(x star) right
17:32and the width is well however much i chopped it up right so if i put n rectangles here and i make them all equal so if i used wanna use a number if i used 100 rectangles how wide nobody knows?
yeah? you in the back
18:00well kind of youre close youre real close, yeah?
yeah. its not a+b/100 because that would say this distance plus that distance over 100 instead its this distance which is this-that over 100 so youre close but not quite she knows all the stuff so she can leave if you know all the stuff you can leave too
18:33and if instead of 100 rectangles we use n rectangles then its (b-a)/n alright?
now let me let me point out 1 thing here umm about how this is working i dont want you to memorize all sorts of formulas and stuff i want you to think about whats going on if you try and do this class by remembering
19:01you either have to have an amazing memory or youre going to screw up and even if you have an amazing memory next year or the year after you will have forgotten if you think about what the idea is and try and see how the formulas fit together with the idea this class is not so hard well there are easier classes but thats the way to succeed is understand whats going on
19:31and then the memory becomes easy because the memory is oh its kind of like that oh its this because thats how it works but if you just memorize formula youll make little mistakes like (a+b)/100 rather than (b-a)/100 or youll remember a wrong formula and youll screw up so this is the width, fine
20:00so now oh good that one works umm so now if we want to find this area well the area with 100 rectangles is going to be well we take 100 points scattered along there let me just call them
20:30so so lets call them x1, x2, x3, blah blah blah im going to forget about the star up to x100 soon im going to change the 100 to an n is going to be well i take the height f(x1) and i take the width which im going to write out here (b-a)/100 and i multiply them together and then i take the next guy
21:02f of.. f(x2) times (b-a)/100 plus f(x3) blah blah blah just go im not going to write all 100 of them and then i get to the last one and i mulitply it by the width and thats what i get so this is my approximate area its not the actual area but its pretty darn close
21:30because theres 100 little rectangles there theyre all pretty skinny its really close this is tedious to write most of you are probably familiar with sigma notation anyone not familiar with sum notation?
youre not familiar? okay so i will now introduce it here we're going to use this notation a lot
22:01but probably not until october we'll use it a little now and then we'll use it a lot in october so this is tedious to write and if instead of 100 i had 1,000 itd be even worse and if its n i wouldnt even know what to write so but we're adding a bunch of things and they all look kind of the same except for 1 little change so this means this is a greek capital S
22:31standing for sum and the things we're going to add up, well we're going to add it up little n things, let me use i because i used n somewhere else starting from 1 and here going to 100 and what are we going to add up? we're going to add up f(x of i) so this notation means exactly what is in the parentheses so these are exactly the same thing
23:01theres no math here theres only notation umm this is exactly the same way to write that so this big sigma shouldnt be scary if you program this is a 4 loop a 4 loop with a sum so this is for i from 1 to 100 umm what line should i write it how about i dont know basic do sum equals f of of..equals old sum plus f(x sub i)
23:36so this is just take this thing add it take the next thing add it, take the next thing, add it we will study these in much greater detail in about 6 weeks, 4 weeks some number of future times okay so thats the area of the 100 rectangles if we change it to 1,000 rectangles it doesnt look a whole lot different
24:04what? oh thank you theres another 0 i missed i'll do that kind of thing a lot umm its pretty much the same so in general the area is going to look like (b-a)/n where n is a hundred, a thousand, a million, 5, whatever you like lets put the sum on the outside
24:31i=1 to big N here n is a thousand times f(x sub i) now we havent used this idea of limit yet im sort of at a picture at this level the area is very close to something straight but its not exactly equal to something straight so now
25:01the thing we do is.. instead of taking the limit well we take 2 limits sort of at once the rectangles get small and the number of them gets big so the area actually we know that the area is going to be just about
25:30(b-a)/n times f(xi) sum i=1 to n and we can show and im going to skip over that that is going to be less than the sum of this form depending on how we choose the xi and bigger than another one and these two things go together let me draw it in a picture
26:02so when we do n=4 heres my function we want to go from there to there i want the area im going to chop it into 4 pieces there we go and i can certainly find i didnt tell you how to choose these xi's except that they live inside this region of something so i get to choose x1 in here somewhere if i choose it here
26:31this is x1 and then i choose x2 well, looks to me like i went below this here this is my x2 and then im going to choose it here this is my x3 and then im going to choose it looks like about here theres my x4 then the area i will get
27:00will be definitely too small but for sure its absolutely too small it can be no bigger i mean no smaller because everywhere the little rectangles, or the big rectangles in this picture sit underneath the curve so this is a lower sum so this is a lower bound
27:37the area is for sure even if i know nothing else about the function except for the value here here here and here the area is for sure less than what i get for those 4 rectangles bigger those four rectangles are less than what id get for the area i want but i can play a different game let me just draw the same picture well pretty close to the same picture
28:03uhh here here here there so if instead i chose maybe maybe instead of calling them x ill call them i dont want to call them y v im going to call them m m for max and min whatever, i would call them v if i instead choose this point
28:32to be my first one for this rectangle and then i choose well it looks all the same but this point to be my v2 here and then i choose this guy to be my v3 and then i choose this guy to be my v4
29:00then i get something that is for sure too big right?
now this should be review to all of you but if you forgot it, whatever so that means that so that means that the area is between and maybe its equal but probably not
29:33the sum i=1 i use 4 of.. i didnt give these numbers but b..a... (b-a)/4 times f(xi) and (b-a)/4 f(vi)
30:02the actual area is between those 2 numbers does everybody agree?
no matter what the function looks like as long as i choose those according to the rule i chose the xi's are the minimums and the vi's are the maximums ok but theres nothing magic about 4 could have done the same thing with 40
30:30or 400 or four thousand or four million or just some number n and i would still get the areas trapped between those 2 things so now the magic trick is..but notice also that if i use lots and lots of rectangles and the function isnt too crazy
31:00this will get bigger if i use twice as many rectangles if i use twice as many rectangles i get a taller one there and i get a taller one here and i get a taller one here and i get a taller one over here so i would get a bigger area on the bottom and a smaller area on the top and so as we take more and more area-uhh rectangles these 2 numbers go to the same thing
31:31so if we take the limit as n goes to infinity lets do this one of either of those or any other way of choosing them then just goes to some number so if this number exists this is the area
32:06and we call this area the integral so the integral is this area
32:33right the integral is this area so this is the definition of the integral if we chop it up into a bunch of rectangles choose a point inside the rectangle and take the limit as the number of rectangles goes to infinity and the width goes to 0 this gives us a thing called the integral and its the area
33:03so why did i go over this because youve already covered this before this understanding that when were doing these integrals comes down to looking at little slices its extremely important for understanding what we do later so even though just like when we did derivatives after awhile you stop using this formula you think back to when you did calculus 1 or calculus a or whatever you call it
33:31you used this formula for a couple of weeks or a month or whatever and then your professor showed you a magic trick that you never need this formula anymore and he said geez why did i sweat so hard to do that formula and similarly when you learned about integrals they probably made you add up little rectangles over and over and over and you were like god i hate this stuff raawwwrrr
34:00and then magic happened and theres an easy formula what we like to do in math is.. take a hard problem understand it fully and then make it easy if we start with the easy then when we try to apply it to something else where the original problem doesnt fit its not going to work you dont know how to adapt it so its important and we keep emphasizing
34:30this definition because this is whats really happening its like.. if you have a car theres a motor in there if you have an airplane theres a motor in there the same principle..well kind of maybe not an airplane how about a boat the same principle that makes the boat motor and car motor run is the same you have no idea whats going on in the motor because you'll think that a boat motor
35:01driving a boat and driving a car have nothing in common well boats might have sails and so bad analogy but the motor or yeah the thing that makes the motor go is the same theres little explosions going on and this things going up and down and that sort of thing this is the motor that makes it work and we want to change this to make our motor not calculate areas but calculate something else you need to understand thats whats happening
35:33so we want to emphasize this because if i just start with which you can probably all do whats the integral is x^2 we all know that its 1/2x^3 plus.. i think you got that 1/3!
you all know how to do that i hope umm so you all know how to do this thing
36:02but some of you have probably forgotten this part or just remembered got it was some horrible thing that they made us do and i hated it thats why im in 132 instead of 126 youre not gonna make me do this umm so just reminding you so that i dont have to make you do it until later umm okay so given this business without even knowing yet how to calculate integrals
36:32we know some stuff about it because integrals are areas well one observation that we can make here from this picture is.. so the way we define this only works if the function is positive if the function is negative its not really an area right if my function
37:02goes like this then we like to think of this area as being a positive number if we ask what is the area of.. how, you know, wanna know the area cause im gonna put carpet over this thing i dont want to buy 0 square ft. of carpet but the integral if i drew it
37:30pretty close if i drew it right it is certainly possible so so it could be true that the integral from 0 to 5 of this function f(x)
38:02is 0 if you use the formula you may come up with 0 it may be 0 but the area is not 0 thats because if you think about this definition in this part your rectangles
38:30have a positive height but in this part your rectangles have a negative height because the formula does not say take the absolute value of the function which is the distance from the graph to the axis but just take the value of the function
39:00so if we wanted area we have to change the sign here but if we want integral then we think of this as negative and we think of this as positive. so integral by this definition without putting little absolute values there is not actually an area its assigned area and thats actually and that may seem like a defect and it is a defect
39:30from the original design but its easily remedied the problem is to discover how much carpet i need to cover this area but the integral is more than just a way of finding area its a way of averaging a way of finding distances a way of finding all sorts of things and we could easily correct it by just taking absolute values
40:00so its not a defect that something with a positive area in the sense of carpeting has a 0 input right?
uhh ok and those i didnt even tell you how to calculate anything its just built into the problem that this is gonna be negative cause its below and the way that we do it, its negative umm and we can do some simple integrals without
40:47in some cases, and you probably did on the homework assignment there are some of these things you can calculate the integral without being able-without having to calculate the limit for example
41:00lets take a specific function f(x)= absolute value of x the graph of the absolute value of x looks like this so lets make this be a clicker question the integral from -1 to 1 of the absolute value of x dx is..
41:30a... uhh...2 b) 0 3) 1 uh thats not a c c) 1 d) -1 how about e.. uhh you need some formula
42:00okay so this is a ok i hope the right answer is there too uhh well your clicker says 1A so put it by letters or
42:30do not if your answer is c do not click the button marked 1 if your answer is c hit the button marked 3 which should also say 1/c yes it does so do it by letters cause theres no number called dunno
43:00anybody need more time?
okay so im going to stop this now last chance, then we go oh come on you didnt answer when i said do you need time uhh too bad ok so it seems that 14% of you think this is the right answer
43:30i could show you the graph but i wont 12% of you think this is the right answer 75% think thats the right answer everybody knows so most of you think that its c and most of you are right its unfortunate for those of you that thinks its those but you just look at this picture this is a question about area
44:01what is the area of this triangle?
1/2 the height here is 1 because the absolute value of 1 is 1 the width is 1 so this area is 1/2 over here this graph is above the axis so its area is also 1/2 you could also think cut this out lay it down there you get a 1x1 square if the area of a 1x1 square is not 1
44:30something is very wrong so the answer is in fact 1 but dont worry you'll have more chances umm soo here let me give you another chance we have another few minutes umm none of these are the questions i prepared, cool uhh so suppose we do the same thing
45:02instead of absolute value of x lets just do the integral from 0 to 1, no not 0 to 1 -1 to 1 of x dx i dont have to draw the picture im told uhh the choices are a)2 b) 0 c) 1 d) -1
45:31e) cant say and f) a rabbit so we have at least 1 bold person who answered rabbit that person is either bold or doesnt know how to work their clicker or his clicker i dont know
everybodys answered?
or just pick the rabbit oh more people somebody just changed their mind change it right back come on umm okay im stopping this now if you are no longer bold and want to-there that person put it back good for you alriiiight youre stuck with it now okay so the correct answer favored by 84% of you
46:33is in fact 0 because the graph y=x blah blah blah the graph of y=x looks like that this stuff is plus this stuff is minus we're going from there to there they cancel out we get 0 uhh another property or 2 that we can use
47:01is.. we can break up integrals because theyre areas if we wanted to find if we wanted to find the area
47:30of the function i use we wanted to find the area of that thing we can integrate from here to here and then integrate from here to here so this is a this is b and this is c and this is f(x) then for sure the integral of f(x) from a to c is the same thing as the integral from a to b
48:00plus.. the integral from b to c right we can break up integrals umm obviously if its with a 0 the integrals gotta be 0 so so something like..
48:40the integral from 3 to 3 sin... (pi x^2) oh this should be pi x^3 dx uh no 3 what is this?
better be 0 doesnt matter what function i put here
49:00because we're saying okay we have this function sin pi (x) and we're 3pi so we must be here umm and we're saying what is the area of that line the area of that line is 0 because it has no width and this is the area of a line one last thing and then i guess we're out of time
49:32is this has an ordering we always put the smaller number on the bottom and the bigger number on the top if we switch them it changes the sign we go backwards so once in awhile we end up going backwards and we'll do some of this going backwards later so the integral from a to b is the same thing as negative the integral from b to a
50:03so i will continue and hope to finish this review on wednesday we have no class on monday because its labor day the homework due a week from today