| Start | integral stands for a bunch of things but mostly
in this context its going to be area under a curve.
you have a curve blah blah blah from a to b so we learned how to do all the approximating stuff. now if you actually want to find the area you do the integral from a to b of f(x) dx where thats f(x). |
| 0:31 | i did this the other day.
this integral so thats a giant s which stands for sum cause thats essentially doing the riemann sum. theres a seat right there if you want. you sure? we could all watch you walk across. itd be an exercise in awkwardness. so this is the antiderivative of f evaluated at b minus antiderivative evaluated A so a capital F would be antiderivative so working backwards. |
| 1:00 | so for example
if i wanted to find
whats a good example?
f(x) =(1/x)^2 from 1 to 3. and i want to know what is that area? why do i care what that area is? cause we want to know what the area is. |
| 1:35 | that.
so that area and now i have to take the antiderivative of (1/x)^2 well thats the same thing as x^-2 which is x^-1/-1. i use that little line to symbolize im going to plug in 3 im going to plug in 1 and im going to subtract. |
| 2:05 | so remember x^-1
is 1/x so this is -1/x
from 1 to 3.
-1/3-(-1) for 2/3. |
| 2:32 | so thats what the integral is all about.
and this is integral calculus and you are going to spend basically at least half the course just learning how to do integrals. remember the integral stands for the area so we could ask you stuff where you cant actually do the integration but its just fun with areas. for example |
| 3:05 | i give you something annoying like this.
say that is 1..2..3..4.. 5 whatever. uhh 6. 1..2..3..4. |
| 3:30 | sorry about the scaling in there.
okay say you have some set of lines like that it makes some obnoxious shape and i say what is the integral from 0 to 4? so this is geometry class. |
| 4:01 | so if i said what is the area-the integral from 0 to 4
im looking for this area.
you cant integrate that function its a weird function. but of course you dont need to integrate it. you want the area from 0 to 2 its going to be a triangle. thats a trapezoid and thats a rectangle. so lets see i would have from 0 to 2 would be 1/2 base times height. |
| 4:32 | plus the trapezoid which would be 1/2
sum of the bases
times the height.
base of 3 and base of 2 you could of course break that up into triangle and rectangle. and the rectangle which is just base times height. lets see thats 3 + 5/2+2 which is 15/2. so if i said whats the integral from 0 to 4 |
| 5:01 | its just 15/2.
so far so good? course i could make it harder. what if i wanted to find integral 2 to 8? of f(x) dx..take a minute and figure that out. |
| 5:40 | thats an 8 if you cant read my handwriting.
the integral from 2 to 8 what were going to do is were just going to add up all these different areas. so lets see. from 2 to 3 i did that one before. thats 1/2 |
| 6:02 | sum of the bases
times the height cause this is a trapezoid.
now of course you could cut it right here and make it a triangle and a rectangle if you wanted. now i have this rectangle. which is the base of 1 and the height of 2. then i have this trapezoid. base of 1 height of 2 and a height of 1. |
| 6:38 | then i have
lets see thats a trapezod
base of 1 and a height of 1 and 4.
and a triangle. which is a height of 4 and base of 2. |
| 7:03 | ok? so again
so first i got
trapezoid
so make sure you know the area of a trapezoid right?
its 1/2 sum of the bases times the height. ok? if you forgot that. area of the trapezoid you add the two bases times the height. so you got trapezoid here got a rectangle here |
| 7:30 | and another trapezoid
another trapezoid
and a triangle.
and you add that up and you get uhh 5/2 +2 +3/2 5/2 +4 which is 5 7, 11 12.5 or 25/2. ok? anybody get 25/2? yess you two get a's i dont even know who you are. anybody else? |
| 8:00 | alright you all get a's.
so now of course how do we make this more entertaining? ill leave that for a minute and make sure everyones got it. so you understand what i did so when im doing the integral all im doing is doing the area starting here finishing there. and its just the area under the curve so most of the time you dont get nice easy lines. and you cant just divide it up geometrically you have to actually come up with something complicated. |
| 8:30 | thats why you know you take this class cause you guys could all do triangles and rectangles.
you could always cut them out and paste them and measure them with your rulers. what if i give you a slightly more annoying picture? something like that? |
| 9:15 | well now notice
the area above the curve is going to be a positive number
the area below the curve is going to be a negative number.
ok? so if you were finding the area you get up to here |
| 9:30 | you found some area and now youd be subtracting.
from here. ok? pull out -2. then you start adding again and youve got that area. that make sense? so if i wanted to find the area from 0 to 3 f(x)dx very soon we learn what to do with that dx. |
| 10:01 | well
the area of this rectangle
that trapezoid and that triangle.
so this is 1/2(1x2) i lied this is 1x2 its a rectangle. 1/2 height of 1 and the bases are 2+4 and the triangle 1/2 of 1x4. |
| 10:34 | k and thats uhh
2+3+2=7.
so the integral from 0 to 3 is going to come out 7. but now what if i want to do the integral from 0 to 5? well we have a couple things we already know from 0 to 3. |
| 11:03 | ok?
so one thing you could do is you can say its going to be the integral from 0 to 3 plus the integral from 3 to 5. and you already know the integral from 0 to 3. so notice you can add those 2 integrals together now what is it from 3 to 5 well thats just |
| 11:31 | base of 2 and a height of 2
so 1/2 of 2x2 is 2.
but its a negative number so the area is 5. somebody looks puzzled. so you understand why you subtract? the area is below the x axis so area above the x axis is considered positive. area below the x axis is considered negative. we found the area up to 3 and we said thats equal to 7. and now youre going to subtract |
| 12:01 | the area of that triangle.
the area of a triangle is 1/2(base x height) 1/2 of 2x2 is just 2. so thats negative 2. then if i went all the way to 7 well this has also got an area of 2. so id be adding that back so i get back up to 7. you could end up with a negative area. if you had a picture |
| 12:37 | say you had something like this and this was i dont know lets make it up lets say thats 8 -20 then uhh 4 |
| 13:00 | then the integral from 0 to 2
would = 8.
the integral from 0 to 10 would equal -12 because of the 8-20. so you can get a negative answer ok? negative number in a physical world, physics world usually just means other direction. so you know for example if this was a velocity curve |
| 13:30 | this area would represent the distance traveled in a particular
well usually time.
so if this was how fast something was going and this was the time the thing hits in 10 minutes you know its gone -12 so its gone 12 to the left instead of the right down instead of up or something like that. okay so negative isnt necessarily bad. if thats your bank balance its bad. thats when you call mom or dad right? dad, mom. whoevers easier. grandma. |
| 14:00 | alright umm
i used to do a lot of begging
you know what the trick is say you have to buy another book for class.
really expensive then you download the pdf. i am posting the homework pages i hope you know that im not dumb. i recommend since im posting the homework pages go out and buy a used previous edition of the book. for like 8 dollars from amazon dont go out and buy a brand new copy of the book thats unneccessary. |
| 14:30 | for those of you who have not already illegally downloaded pdfs.
which i cant tell you to do because thats illegal. we didnt have those when i was in college. we just invented books we used to write on rocks. it was very complicated. you know wed take dinosaur blood and it would dry the rain it was terrible you lost all the stuff. but then we discovered fire it made it easier. so you could get a negative area |
| 15:01 | if you just, you could stop in the middle
you could do all sorts of things so for example
say i ask you, whats the maximum
area?
well at this point youre at 8 then youre subtracting then you get back up to here so the biggest area youre going to get is 8 if this was something where i said whats the maximum umm total distance or something it would occur at x=2. if i said where would the minimum be, the minimum would be here. |
| 15:31 | at x=10.
so we could ask you something annoying like that. how we doin on these so far? if you took the AP they have lots of these on the AP love these questions, i dont know why. so heres a variation of it that i also would concentrate on learning. |
| 16:00 | oh so before i forget so were learning stuff today
we're learning stuff on monday
wednesday
about halfway through is going to turn into a review session for the midterm
all the following monday will be a review for the midterm.
im giving you about 2 hours worth of review. hope thats okay. because im not going to be able to have time for a special review session. so these will be reviews okay? um sorry about that. you can blame the snow gods. |
| 17:01 | ok.
so lets say this curve is f(x). so this is whats called an accumulation function. accumulation as in to accumulate, to add up. ok? what happens-what does this mean? this means find the area from 0 to x. so g(x) means find the area from 0 to x underneath the curve f(x). so in other words you just keep going to the right and you stop at x. |
| 17:32 | so g(5)
would be the area
from 0 to 5
of f(x)dx.
i think you have a homework problem that looks like this. if you dont, you will. so g(5), we-wait ok? because what does g(5) mean? well remember x is g(x) right? |
| 18:01 | you go from 0 to 5
of f(x).
you really want to avoid the confusion change that letter doesnt really matter ok? you could make it t, make it banana the area so that this g tell us what the area is so if i wanted to find |
| 18:30 | g(10)
you have the integral from 0 to 10
thats f(t)
dt.
that would be +8-10 =-2. you understand why thats -2? youre just adding the areas. |
| 19:01 | okay and if i said
whats g(12)?
itd be the integral from 0 to12 of f(t)dt which would be 8 -10 +6 which is 4. so now what if i said what is the maximum, where is the maximum? |
| 19:36 | what value of x is the maximum?
at 5. because thats where you had the biggest positive area now you subtract and even though you add 6 back later you dont end up with as big a number because you subtracted more. if this was 16 then you would end up with a bigger area when you got at 12. does that make sense? ill say that again. so right now you have 8-10 is |
| 20:02 | -2
when you had 6 you get 4
if this was big enough
that looked like that
then you would get this would be where the maximum is.
the minimum would occur at 10. where is g(x) increasing? |
| 20:30 | sure you want to go for it right away?
so lets think back to 125 when a functions increasing what do you know about the derivative? the derivative is positive. if g(x) is the integral of f(t) what do you think the derivative of g(x) is? its just plain old f(t). because the derivative and the integral sort of wipe eachother out |
| 21:00 | we're going to learn more about that in a minute.
or if you just think about the area the areas increasing all the way until you get to 5. then the area starts to shrink. the area would decrease all the way until you get to 10 and then the area would increase again. so its going to be increasing on 0 to 5 and 10 to 12. |
| 21:32 | and decreasing
on 5 to 10.
make sense? whos lost? whos lost? everyone understands that? wonderful! that makes me very happy i dont have to teach it twice. alright so this brings me to something called |
| 22:00 | the fundamental theorem of calculus.
very scary sound. but that has to do with the relationship between derivatives and integrals. we'll come back to more of this. |
| 22:33 | so the fundamental theorem of calculus
sounds so important.
its fundamental. this is one of those things that newton discovered. and newton found that there is actually a relationship between derivatives and integrals. thats the whole antiderivative thing. so we're going to learn how to find the derivative of integrals. so for example |
| 23:00 | lets say i told you that f(x)
integral from 5 and x
of sin
t dt.
and i want to know what is f prime at x? really easy. we're doing the derivative of the integral now so youre not doing the antiderivative. youre just doing the derivative of the integral so youre doing the derivative of the antiderivative. |
| 23:32 | so this just comes out
sin(x).
thats it, no +c. just sin(x). why is that true? well lets lets do this out. first the integral from 5 to x of sin(t)dt. well remember what the antiderivative of sin is its -cos. |
| 24:00 | the derivative of -cos is sin.
we go from 5 to x so that is -cos(x) minus minus so plus cosine of 5. whats the cosine of 5? i have no idea |
| 24:31 | okay but its a number, its a constant.
so what i take now im taking the derivative what is now this is f(x) so f'(x) is take the derivative. whats the derivative of -cos? sin. and whats the derivative of cos of 5? 0 cause cos5 is a constant. so you end up right where-right there. |
| 25:01 | thatll lead us to a rule.
derivative integral from any constant x f(t)dt just f(x) and that is the first fundamental theorem of calculus. |
| 25:38 | here? first i took the antiderivative
then i took the derivative again.
i took the antiderivative and what happened was the bottom number just comes out a constant. so i take the derivative the derivative of that constant is 0 and i just get back where i started. now instead of t i have x. ok? |
| 26:00 | so if i go from any constant
to x
of f(t) i just end up with f(x).
this could be a billion could be negative a billion it doesnt matter. could be high cause i really want to mess with your head k eq really want to mess with you doesnt matter. cause its a constant. and this just means derivative so if i had the derivative integral from 3 to x of e^t^2 dt |
| 26:32 | that just becomes
e^x^2.
notice no constant yes? student: where did the t go? where did the t go it got replaced by x. because here right this is cos(t) from x to 5? so i plug in x for t and i plug in 5 for d. so these are what are called ?? variable, doesnt really matter what the variable is. so it can be very confusing |
| 27:00 | also do not confuse this
with finding the integral
which people often do. here we are finding the derivative
of the integral.
its like doing the square root of a square. or remember all that inverse function stuff we did last semester? this is, youre kind of doing that. ok? fortunately were not doing the whole f inverse f blah blah blah k cause thats hateful. im teaching that to my other course very soon. youre done with that forever |
| 27:30 | and ever.
so thats called the first fundamental theorem of calculus so if you wanted to find from 3 to x of e^t^2 it would be that if this was 30 to x its the same answer. okay if this was -30 still the same answer it doesnt matter now of course i have to make this harder. i could just stop here god forbid. |
| 28:05 | suppose i have f(x)=the integral from 5 to x^3
of sin(t)dt.
its not really much harder just a little bit but you know why i make these harder we dont really like you guys very much thats why. i pretend i do but the truth is i live to torture you. its a happy day when youre suffering and crying. |
| 28:31 | you know and all that its just wonderful.
anyway why do i make it harder? cause otherwise you'd all get A's and we cant have that so this is just what you did before. want to find the derivative-sorry. f'(x) would be you plug in the x^3 and then you use the chain rule. whats the derivative of x^3? 3x^2 so multiply by 3x^2. |
| 29:02 | thats not so bad.
why? you guys want to know why? i want to make sure you understand. lets do a why, okay? i know how important theories are to you guys. as i said theres a little bit of chain rule involved. or we can see by doing it out. same thing from before. you want to do the antiderivative so go ahead and do the integral of sin(t). |
| 29:32 | that was -cos(t).
from 5 to x^3. which is -cos(x^3) + cos5. now im going to take the derivative. whats the derivative of -cos(x^3)? its sin(x^3) times derivative of x^3. so thats where chain rule comes in. |
| 30:07 | so now you sort of
fuller version of the first fundamental theorem of calculus
ill make that a g.
|
| 30:30 | this becomes f evaluated g(x)
times
g'(x).
so again its not really very hard but lets |
| 31:09 | suppose i wanted to find the derivative of that integral
theres a prime here, kind of hard to see.
well you plug in the sin x wherever you see t. and then since sin is a function you multiply by the derivative of sin. thats it. we dont care about the 8. |
| 31:34 | good?
i could make this of course slightly more annoying. say i wanted to do derivative integral of 5 to sinx |
| 32:10 | now what happens?
do i have to get 2 rounds of sin in there? what happens? this would become the derivative of that would be the square root of sin^2x+sin^3x |
| 32:30 | times cosx.
k? how we doin on these? we dont like? why not? its just plugging in and doing a little derivative stuff if youre waiting for this to get worse it doesnt get worse. yes? |
| 33:02 | i cant hear you.
student: can you do that again? surely. im taking the derivative of this so where i see t, i plug in this function. ok? and multiply by the derivative of this. cause its chain rule. okay, all im doing. so thats the rule. you take t replace it with whatever function it is and youre going to multiply it by g'(x). |
| 33:31 | ok?
theres only one more flavor of this thats just slightly harder but not really. suppose i had another function down here. now what do you think i do? im going to erase this. this was actually on my midterm last year, this exact problem. |
| 34:00 | my midterm
which ill post in about a week.
so well make some samples-after weve written the midterm well give you guys samples so nobody gets disappointed. alright so wherever you see t youre going to plug in sinx and then times the derivative of sin. minus now do it to the bottom. plug in cosine |
| 34:30 | times the derivative of cosine
and youre done.
ok? that wasnt so bad. oh we're going to do a couple of these now, you guys ready? i'll give you 3 practice problems. |
| 35:34 | okay.
why dont you try those 3 variations? so this is-each time find f'(x) write that somewhere. just do the derivative of that integral. we're just going from 4 to x. you just plug in x. so now we get 1+x^2. |
| 36:00 | you get that one?
so far so good? medical school? maybe. what if i have x^3? well wherever i see t, i put in x^3. so its the square root of 1+x^3. squared, which you could make 1+x^6 if you wanted to show off but its certainly not necessary. and i always say dont show off unless youre going to get it right. |
| 36:32 | how we doing?
third one youll definitely get into med school. no doubt ill just call them up right now. alright. but wait i forgot something no med school. funeral parlor. they gotta be dead first. then you can do all you want so i forgot to multiply by x^3. there ya go, there ya go, alright. see, dont show off until you get it right. |
| 37:03 | you know that joke about the funeral-never mind.
alright okay. you cant mess them up..alright. so you plug in x^3 for t and you get 1+(x^3)^2 and then you multiply by the derivative of the power. you multiply by the derivative of that. so now for the hard one we have to do it twice thats all. so this becomes the square root of 1+(x^3)^2 |
| 37:33 | times 3x^2
minus
the square root of 1+x^2
squared
times 2x.
ok? happy so far? howd we do on those? question? why is it minus? its the integral at the top minus the one on the bottom. why were we able to do that? what you really do |
| 38:00 | is you pretend that when youre doing the integral from x^2 to x^3
youre really doing the integral from
x^2 to some constant
plus the integral from some constant to x^3.
okay and you chop it up in the middle okay and since the part with the constant goes away thats why youre able to do this. k but thats not important. how do we like these theyre not so bad? theyre really not that hard you just follow the rules. |
| 38:30 | ok?
now what else could we ask you to do? well we could just ask you to find some antiderivatives. i did some of these the last time but ill do it again. |
| 39:02 | lets practice just a bit.
remember how to do that? |
| 39:36 | can you read that?
i'll work on some darker chalk for next time. not making any promises. the integral from 0 to 1 of x^3+4x^2+5 how do we do this? we take the antiderivative. antiderivative of x^3 |
| 40:00 | x^4/4
antiderivative of 4x^2
4x^3/3.
and the antiderivative of 5 is 5x. from 0 to 1. plug in 1, plug in 0 and subtract. this would equal 1/4 +4/3 +5 |
| 40:31 | minus
well everything on the right side is just going to come out 0.
so you get 79/12. do you guys know how to add something like a 1/4+4/3? what you do is multiply 3x1, multiply 4x4 |
| 41:00 | and you put the 4 and the 3 together.
so 3x1 is 3 4x4 is 16 4x3 is 12 so youre going to get 19/12. okay? you never learned that trick before thats the way you combine the fractions. very fast. you want to combine 2/3 and 4/5 thats 10..thats 12 thats 15 you get 22/15. |
| 41:31 | your teachers never teach you that?
thats because youre young. if you were old like me you would have learned that ok? it works every time it also works with subtraction. so if you had 3/8-1/7 its 21-8/56. which is 13/56. okay thats how i do it so fast in my head thats all you have to do. wow, how come we never learned that/ |
| 42:00 | call up your 5th grade teacher and ask if he or she knows it
and theyll probably go i didnt know that.
is that going to be on the test? well this might be on the test ok? and i dont know if professor T expects you to know the fractions or if you can leave it there. i let you leave it there but she might do otherwise. probably agrees with me but you never know. but you should learn this trick anyway its very handy. do it at parties if people have trouble sleeping it works really well. |
| 42:33 | its not so good for the dates but thats okay.
alright lets see if you can do another type of integral. |
| 43:03 | im taking this directly from the book so its slightly nasty.
not my problem. how do we do this one? factor out the half so now we have 1/2 the integral from 1 to 9 of 1/x dx. and whats the integral of 1/x? |
| 43:30 | lnx, lnx.
pronounced log ln of the absolute value of x. from 1 to 9 you have that 1/2 outside. so thats 1/2 ln9 -1/2ln1 and the ln of 1 is 0. so its 1/2ln9 |
| 44:00 | also known as the ln3.
either answer would be fine. you know why its ln3? you put the 1/2 back up and thats the square root of 9 square root of 9 is 3 so you do problems like this in the book and you get to here and you dont understand why they have this they do that just to show you theyre nasty. ok? you guys get the idea this is all fundamental theorem of calculus. |
| 44:35 | this is good enough.
you can stop here you dont need to do that stuff. alright so again you pull out the 1/2 cause remember 1/2x is 1/2 times 1/x. so now you got 1/2 and the antiderivative of 1/x is lnx. you dont really need the absolute value because 1 and 9 are both positive. but you should do it just for practice. |
| 45:03 | back to the good stuff.
oh there is one other little thing i havent shown you guys. its not complicated. its actually very straightforward. |
| 45:50 | alright the integral from a to b of f(x)dx and thats equal to some value k and what about the integral from the other direction |
| 46:00 | if you reverse it-these are called the limits of integration.
instead of going from a to b you go from b to a. what happens to k? you get negative, good you just go the other direction. -k, did i do this already? suppose i gave you the following |
| 47:02 | the integral from 0 to 10
of f(x)dx is 25.
and the integral from 4 to 10 of f(x) is 13. what is the integral from 0 to 4? 12, right? cause all you really do is youd say the integral from 0 to 4 |
| 47:32 | plus the integral from 4 to 10
equals the integral from 0 to 10
ok?
not so bad? so the integral from 0 to 4 is what were looking for the integral from 4 to 10 is 13. the integral from 0 to 10 is 25. |
| 48:02 | the x would just equal 12.
so far so good? i could make that from 4 to 0 instead of 0 to 4. and then your answer would just be -12. we liking this? this is not so bad believe it or not this is the theoretical part its the abstract part. we're going to start to get much more |
| 48:30 | mechanical.
i know youre going to like that better. whenever i see something what do i do? do we have questions on the homework? web assign? there are things i havent covered today. that was the idea we were going to catch everything up. |
| 49:13 | oh right!
thats good im glad you asked that. suppose i get this i put in my notes that i put up on blackboard but obviously for the 3 of you who look at the notes its not a problem but for the 227 of you who did not and i wonder why no one likes me. |
| 49:32 | okay so
if i wanted to find
the limit
as n goes to infinity
of sigma i=1 to n
of i
i^3/n^4.
ok that is an integral. that is going to be the integral |
| 50:01 | from 0 to 1
of x^3dx.
how do i know k well first of all they tell me its on the interval from 0 to 1 okay so that tells me this is on 0,1. this is homework problem 75. okay but you know you have to be able to do all of them so ill just do an example of 1. like this one so the 0 to 1 tells me im going to integrate from 0 to 1. |
| 50:30 | okay this
tells me what my function is, see i^3?
its x^3 with i^4 itd be x^4. with i^3+1^2 x^3+x2. from here you should be able to do the integral on your own. k so if i had |
| 51:16 | that would be
the integral from 0 to 1
x^3+x^2.
|
| 51:31 | well what-this is a riemann sum
ok?
so if you evaluate this its the same as evaluating that. thats what i was sort of showing last wednesday no one in the whole room was getting it. ok? the n goes away cause youre doing the limit. if the limit goes to infinity you evaluate all that youd be doing the same thing as evaluating this integral. okay this is the connection between those sums and integration. |
| 52:01 | when you go to infinity
those sums vanish and turn into the integral symbol.
yes? why? oh cause it says its on the interval 0 to1. you look at the problem. i'd have to give you an interval if it was on 0 to 2 then this would be from 0 to 2. okay but then that would become 2i^3/n^2 so it starts getting messy. so i dont want to overcomplicate things. |
| 52:33 | alright so
study hard
monday well give you some new stuff.
look at what i put on blackboard that could be useful. |