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MATH 16A LECTURE.
PROFESSOR:
DECEMBER 9, 2008.
WELCOME BACK TO THE LAST LECTURE OF THE SEMESTER.
HOPE YOU ALL WILL MISS IT AS MUCH AS I DO.
PLANNING TO DO TODAY WAS FINISH THE BOOK.
AND 6.5.
I
SO WHAT I WAS
FINISH SECTION 6.5
I THINK WILL BE A LITTLE BIT OF TIME LEFT OVER AT THE
END AND YOU CAN HAVE YOUR CHOICE.
WE CAN EITHER DO SOME
PRACTICE FINAL EXAM, THE ONE ON THE WEBSITE.
DO MY REVIEW.
THE ONE I THINK IS MOST IMPORTANT FROM THE SEMESTER BUT THAT WILL
HAPPEN AT THE REVIEW SESSION WHICH WE HAVEN'T SCHEDULED YET OR I
SUPPOSE YOU CAN GO EARLY.
WE WILL VOTE.
SECTION 2.4, ABOUT AREAS.
SO LET'S FINISH.
AND SO LET ME JUST DO A FEW MORE
EXAMPLES JUST TO REMINDS OURSELVES OF WHAT WE WERE DOING THERE.
AND MAYBE I'LL DO A FEW EXAMPLES OF AREAS THAT ARE SLIGHTLY MORE
INTERESTING THAN THE ONES IN THE BOOK.
HOW ABOUT THIS ONE.
THIS ONE IS IN THE BOOK.
SO FROM MINUS ONE TO
TWO.
(ON BOARD).
SO THIS IS SORT OF SIMPLEST KINDS OF EXAMPLE.
THOSE TWO, Y-EQUALS X-SQUARED MINUS TWO X-.
PARABOLA.
HERE.
TWO.
IF I DRAW
GIVES ME A
MINUS E-TO THE X, GIVES ME THIS CURVE GOING DOWN
THEY NEVER CROSS.
I WANT THE AREA BETWEEN MINUS ONE AND
SO IF THIS AREA RIGHT HERE, AND SINCE THOSE CURVES NEVER
CROSS IT'S VERY EASY TO SAY, IF YOU WRITE DOWN THE RIEMANN SUM,
IT'S GOING TO BE THE AREA OF THAT RECTANGLE PLUS THE AREA OF THAT
RECTANGLE PLUS THE AREA OF THAT RECTANGLE, SO YOU HAVE TO KNOW
HOW TALL THAT RECTANGLE.
INSTEAD OF IT BEING THIS POINTS,
SUBTRACT THAT POINT -- SO THE AREA IS GOING TO BE THE INTEGRAL OF
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MINUS ONE TO TWO OF THE FUNCTION THAT SAYS HOW FAR APART THOSE
TWO CURVES ARE, AND THAT FUNCTION IS SIMPLY TAKE THE TWO THINGS
AND SUBTRACT THEM.
THAT'S JUST THE DISTANCE FROM THE TOP CURVE
TO THE BOTTOM CURVE.
AND SO WE CAN INTEGRATE THAT THING. TERM
BY TERM SO THE INTEGRAL OF X-SQUARED IS X-CUBED OVER THREE.
INTEGRAL OF MINUS TWO X-IS MINUS X-SQUARED.
MINUS JUST TURNS INTO PLUS.
SO IT'S PLUS E-TO THE X-AND
INTEGRAL OF THAT IS PLUS E-TO THE X-.
INTEGRATION.
AND THAT MINUS
SO I PLUG IN TWO.
SO JUST TERM BY TERM
FIGURE IT OUT.
PLUG IN
NEGATIVE ONE, FIGURE IT OUT AND SUBTRACT AND IT COME OUT TO ABOUT
SEVEN.
A LITTLE BIT MORE.
SO THAT'S THE SIMPLEST CASE, WHEN
THESE TWO CURVES DO NOT INTERSECT.
THEN YOU JUST SUBTRACT THE
TOP FROM THE BOTTOM AND INTEGRATE.
AND IT'S EASY TO FIND THE
AREA IN BETWEEN.
DO INTERSECT.
WHAT GETS A LITTLE MORE COMPLICATED IS IF THEY
SO LET'S RACHET IT UP A LITTLE BIT, ANY QUESTIONS
ABOUT THAT ONE?
LET ME FIND THE AREA BOUNDED BY TWO CURVES.
Y-EQUALS
X-SQUARED PLUS TWO X-PLUS THREE. AND Y-EQUALS TWO X-PLUS FOUR.
NOT GOING TO TELL YOU FROM WHAT A-AND B-ARE.
WHAT THE LIMIT OF
INTEGRATION ARE, YOU HAVE TO FIGURE THAT OUT FOR YOURSELF.
WHAT
WE'RE GOING TO DO IS DRAW BOTH CURVES AND FIGURE OUT WHAT THIS
MEANS, Y-EQUALS TWO X-PLUS FOUR, THERE'S A STRAIGHT LINE, LOOK
SOMETHING LIKE THAT. THE OTHER CURVE IS A PARABOLA.
DRAW THAT PARABOLA.
IT LOOKS SOMETHING LIKE THIS.
SO LET ME
SO WHAT I
SAY WHAT IS THE AREA BOUNDED BY THOSE TWO CURVES, GEOMETRICALLY
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IS THIS AREA IN HERE.
CURVE.
THAT'S THE ONLY AREA BOUNDED BY THOSE
THAT GOES ON FOREVER, THIS IS WHAT IS BOUNDS BY THOSE
TWO CURVES, THE AREA IN BETWEEN.
OUT WHAT A-IS.
A-TO B-.
AND WHAT B-IS.
SO WHAT I HAVE TO DO IS FIGURE
AND THEN DO THE INTEGRAL FROM
BUT I'M NOT TELLING YOU WHAT A-AND B-ARE U-YOU HAVE TO
FIGURE THEM OUT FOUR YOURSELF.
YOU SAY WHERE DOES THIS STRAIGHT
LINE INTERSECT THAT PARABOLA.
SO YOU FIND, SO THE AREA IS GOING
TO BE THE INTEGRAL OF THE DISTANCE FROM THE TOP CURVE WHICH IS
TWO X-PLUS FOUR DOWN TO THE BOTTOM CURVE.
PATTERN AS BEFORE.
SO THAT'S THE SAME
DISTANCE FROM THE TOP TO THE BOTTOM BUT I
DON'T KNOW WHAT A-AND B-ARE.
WHAT YOU'RE GOING TO DO IS FIND
A-AND B-BY FINDING WHERE THE CURVES INTERSECT.
SO FINDING WHERE
TWO X-PLUS FOUR IS X-SQUARED PLUS TWO X-PLUS THREE.
SOLVE QUADRATIC EQUATION TO DO THIS.
YOU HAVE TO
AND THE ANSWER TURNS OUT
TO BE A-EQUALS MINUS ONE AND B-EQUALS PLUS ONE.
WHEN YOU JUST,
SO YOU SOLVE YOUR QUADRATIC, THAT'S YOUR A-MINUS ONE AND THAT'S
TURNS OUT TO BE B-PLUS ONE.
SO THE AREA FINALLY EQUALS THE
INTEGRAL FROM MINUS ONE TO PLUS ONE OF NEGATIVE X-SQUARED PLUS
TWO X-MINUS TWO X-PLUS FOUR MINUS THREE, AND SO INTEGRATING TERM
BY TERM BEING WHAT THE THE INTEGRAL OF MINUS X-SQUARED, ANYBODY.
PROFESSOR:
IT'S GOING TO BE MINUS ONE THIRD X-CUBED AND THE
INTEGRAL OF ONE IS X-.
AND SO NOW YOU JUST EVALUATE IT AT ONE.
EVALUATE AT MINUS ONE AND IT ALL TURNS INTO FOUR THIRDS IS THE
AREA.
SO THIS IS STILL, SO THIS PROBLEM YOU HAD TO, YOU DIDN'T
HAVE ALL THE DATA BUT YOU KNEW ENOUGH TO DRAW THE PICTURE AND SAY
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I HAVE TO FIGURE OUT WHAT A-AND B-IS THE, THE LIMIT OF
INTEGRATION.
LET ME JUST DO ONE MORE.
LET ME DO A LITTLE BIT MORE INTERESTING.
THE AREA BOUNDED BY, TWO OTHER CURVES.
Y-EQUALS X-CUBED MINUS THREE X-.
BEFORE.
HOW ABOUT FIND
ANOTHER PARABOLA.
SO LET ME DOT SAME THING AS
I'M NOT TELLING YOU WHAT A-AND B-ARE.
TWO PICTURE.
AND
LET ME DRAW THE
Y-EQUALS TWO X-SQUARED S-HERE'S A PARABOLA, SO
HERE'S IS TWO X-SQUARED.
AND WHAT'S THE OTHER CURVE LOOK LIKE?
IT'S CUBIC, AND IT LOOKS LIKE THIS.
THERE'S THE CUBIC.
IT
GOES THROUGH THE ORIGIN THERE BECAUSE WHEN X-IS ZERO, THAT'S
ZERO.
NOW YOU SEE, HERE'S THE REGION WHOSE AREA I WANT.
ARE TWO SORTS OF REGIONS THAT MEET AT THAT POINT.
FIGURE THAT OUT.
I WANT TO
I WANT TO FIGURE OUT THE AREA OF UNION OF
THOSE TWO REGIONS.
FOR.
THERE
THAT'S WHAT THE PROBLEM IS REALLY ASKING
WHAT I INSIDE TO DO IS FIGURE OUT WHERE THAT POINT IS.
THAT'S ZERO AND I NEED TO FIGURE OUT WHERE THAT POINT IS. NOW
WHAT IS, SO WHAT I'M GOING TO HAVE HERE, SO ON, I'M GOING TO GET
TWO INTEGRALS.
WHY DO I NEED TWO INTEGRALS.
BECAUSE HERE THE
TOP CURVE, DISTANCE FROM THE TOP TO THE BOTTOM, WHAT GOES THE
DISTANCE FROM THE TOP TO THE BOTTOM.
MINUS THREE X-.
THE TOP CURVE IS X-CUBED
THE BOTTOM IS TWO X-SQUARED.
THIS A-AND THAT B-.
SO LET ME CALL
SO THE AREA I'M GOING TO HAVE TWO AREAS,
THE AREA OF A-PLUS AREA OF B-FOR THESE TWO SEPARATE REGIONS.
SO
WHAT IS THE AREA OF A-GOING TO BE?
FROM A-TO ZERO.
IT'S GOING TO BE INTEGRAL
I HAVE FIGURE OUT LITTLE A-.
WHAT'S THE
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DISTANCE FROM THE TOP TO THE BOTTOM ?
MINUS THREE X-.
THE ONE ON TOP IS X-CUBED
THE ONE ON THE BOTTOM IS TWO X-SQUARED.
THERE'S THE AREA OF CAPITAL A-.
WHAT IS THE AREA OF CAPITAL B?
IT'S GOING TO BE INTEGRAL FROM ZERO TO LITTLE B-.
WHAT'S THE DISTANCE FROM THE TOP TO THE BOTTOM?
TOP THEY'VE CHANGED PLACES.
HAVE TO DO IT THIS WAY.
FINE.
AND NOW
THE ONE ON THE
NOW TWO X-SQUARED IS ON TOP.
SO I
BECAUSE THIS IS THE NEGATIVE OF THAT.
BUT IN BOTH CASES THIS INTEGRAL IS ALWAYS GREATER THAN OR EQUAL
TO ZERO IF YOU'RE BETWEEN ZERO AND LITTLE B-AND THIS IS GREATER
THAN OR EQUAL TO ZERO IF BETWEEN LITTLE A-AND B-.
SETUP THAT WAY SO ADDING TWO POSITIVE THINGS.
YOU HAVE TO
SO JUST REMAINS
TO FIND A-AND B-BY FIGURING OUT WHERE THESE CURVES INTERSECT.
SO I HAVE TO SET TWO X-SQUARED EQUAL TO X-CUBED MINUS THREE X-.
AND SOLVE FOR X-.
STUDENT:
THAT'S STILL SUPPOSED TO BE TWO X-CUBED.
PROFESSOR:
X-SQUARED.
LET ME WRITE THIS WAY.
MINUS THREE X-.
THANK YOU.
SO I NEED TO SOLVE THIS.
ZERO EQUALS X-CUBED MINUS TWO X-SQUARED
THAT'S NOT A QUADRATIC, IS IT?
BUT WE STILL
KNOW HOW TO SOLVE IT, WHAT CAN WE FACTOR OUT OF THIS RIGHT AWAY?
AN X-.
SO I GET, I CAN WRITE IT THIS WAY.
IT'S ZERO AT ZERO, THAT'S THIS POINT.
CURVES CROSS.
AND NOW I CAN SEE
WHEN X-EQUALS ZERO THE
AND THIS IS NOW A QUADRATIC AND I CAN SOLVE THAT,
I GET X-MINUS ONE X, SORRY, PLUS ONE, MINUS THREE.
THAT'S RIGHT.
(ON BOARD).
SO THAT TELLS ME THIS POINTS HERE IS MINUS ONE,
AND THAT POINTS IS THREE.
SO WHEN I'M FINALLY DONE I PLUG IN
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A-EQUALS MINUS ONE.
AND LITTLE B-EQUALING PLUS THREE, NOW YOU
CAN DO THOSE TWO INTEGRALS.
WILL LET YOU DO THE END.
WHETHER IT'S ALL SAID AND DONE.
AT THIS POINT IT'S SET UP.
I
SOMEBODY
ASKED ME ON THE FINAL IF I ASK A QUESTION LIKE THIS HOW FAR DO
YOU HAVE TO GO TO GET CREDIT, THE ANSWER IS IF YOU GET THIS FAR
YOU WILL GET ALMOST ALL THE THE POINTS.
IF YOU GO AHEAD AND
FINISH IT AND DO ALL THE THE ARITHMETIC TO GET THE FINAL ANSWER
YOU'LL GET A FEW MORE POINTS BUT THIS WILL GET MOST CREDIT
BECAUSE THAT MEANS YOU KNOW HOW TO GET THAT.
STUDENT:
PROFESSOR:
FOR SKETCHING, WHOLE TAKE THE FIRST DERIVATIVE.
ALL YOU NEED TO DO HERE IS FIGURE OUT WHICH ONE IS ON
TOP OF WHICH OTHER ONE AND WHERE THEY CROSS SO YOU CAN FIGURE OUT
THE INTEGRATION.
YOU DON'T HAVE TO DO ANYTHING FANCY.
SOMEBODY DID CAN ABOUT ONE OTHER PROBLEM.
WHAT IF I HAVE THE
AREA, IF I WANT THE AREA BOUNDED BY THREE CURVES.
HAVE A CURVE THAT LOOKS LIKE THIS.
HAVE A CURVE THAT LOOKS LIKE THIS.
A CURVE THAT LOOKS LIKE THIS.
IMAGINE THAT.
CURVES.
SO YOU MIGHT
THERE'S F-OF X-.
G-OF X-.
YOU MIGHT
AND YOU MIGHT HAVE
I'LL CALL H-OF X-.
YOU CAN
AND WHEN I SAY THE AREA BOUNDED BY THE THREE
WELL, HERE IT IS.
YOU MIGHT WANT THAT AREA.
THE SAME IDEA THAT WE'VE DONE BEFORE.
INTO THINGS WHERE WE CAN INTEGRATE.
SO IT'S
YOU HAVE TO BREAK IT UP
SO HERE LET ME DRAW A
VERTICAL LINE THERE.
CALL THAT C-.
LET ME CALL THAT A-AND CALL THAT B-AND
SO I'M GOING TO GET ONE FORMULA FOR THIS AREA.
BECAUSE THE TOP IS F-.
THE BOTTOM IS H-.
AND I'LL GET A
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DIFFERENT FORMULA FOR THIS AREA BECAUSE HERE THE TOP IS H-.
AND
THE, SORRY, THE TOP IS F-AND BOTTOM IS G-AND TOP IT H-AND GOT IS
G-.
-- IF I CALL THIS BA, CAPITAL A-CAPITAL B, THE AREA IS
GOING TO BE THE AREA OF THIS PART, PLUS THE AREA OF THE OTHER
PART.
AND I'M GOING TO HAVE TO WRITE DOWN TWO INTEGRALS AGAIN.
ONE INTEGRAL IS GOING TO BE FROM LITTLE A-IT LITTLE B-.
GOING TO BE THE DISTANCE FROM THE TOP TO THE BOTTOM.
IS LITTLE F-.
THE BOTTOM IS LITTLE G-.
IT'S
SO THE TOP
AND THEN I'M GOING TO
HAVE A DIFFERENT INTEGRAL FOR THIS AREA HERE.
CAPITAL B-.
AND
THAT'S GOING TO BE THE INTEGRAL FROM THERE TO THERE, LITTLE B-TO
LITTLE C-.
STILL G-.
NOW THE TOP IS THIS CURVE.
H-.
AND THE BOTTOM IS
AND SO WELL, NOW YOU CAN GO AHEAD AND DO IT.
DEPENDING ON WHAT THE ACTUAL FORMULAS ARE FOR F-G-AND H-.
DRAWING THE PICTURE WILL TELL YOU WHAT YOU INSIDE TO DO.
SO
TO
WRITE IT DOWN.
SO LET ME DO ANOTHER EXAMPLE.
A FEW MORE EASY EXAMPLES.
BECAUSE ACTUALLY MAKE SENSE SOMETIMES TO ASK FOR THE AREA OF AN
INFINITE REGION.
CURVE.
SO LET ME SAY WHAT I MEAN.
E-TO THE MINUS X-.
THE AREA UNDER THIS CURVE.
IT GOES ON FOREVER.
NICE SIMPLE CURVE.
SO LET ME DRAW A
SUPPOSE I WANT
WHEN DO I STOP DRAWING RECTANGLES,
THE AREA UNDER THIS CURVE GOES ON AND ON
FOREVER.
WELL WE DON'T KNOW HOW TO DO THAT YET.
IT LITTLE B-.
SO HELP STOP
OUT TO B-.
I WANT TO FIND THE AREA FROM ZERO, A-EQUALS ZERO
WE CAN DO THAT, THAT'S A NICE SIMPLE ONE.
SO LET
ME WRITE DOWN THE AREA, INTEGRAL FROM ZERO TO B, OF E-TO THE
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MINUS X-D-X-.
THIS IS EASE I.
IT'S MINUS E-TO THE MINUS
X-FROM ZERO TO B-EQUALS MINUS E-TO THE MINUS B-MINUS MINUS 11
MINUS E-TO THE MINUS B-.
FINE, THAT IS THE AREA.
FROM ZERO OUT TO THIS NUMBER WHICHEVER ONE I PICK.
(ON BOARD).
NOW THE
QUESTION IS WHAT HAPPENS AS I MOVE THIS OUT TO INFINITY.
THE LIMIT.
THAT'S THE QUESTION, WHAT IS THE AREA UNDER THIS
ENTIRE CURVE ALL THE WAY OUT TO INFINITY.
REGION.
TAKE
IT'S AN INFINITE
SO THE AREA UNDER E-TO THE MINUS X-FROM ZERO TO
INFINITE IS JUST THE LIMIT OF THIS THING. JUST TAKE THE LIMIT AS
E-GOES TO INFINITY OF ONE MINUS E-TO THE MINUS B-.
SO WHAT
HAPPENS TO E-TO THE MINUS B-AS B-GOES TO INFINITY.
IT'S THE
HEIGHT OF THIS CURVE.
IT GOES TO ZERO.
YOU GET ONE.
SO THE
AREA UNDER THIS CURVE, IF YOU GO ALL THE WAY OUT TO INFINITY IS
JUST ONE.
IT MAKES PERFECT SENSE TO ASK WHAT'S THE AREA UNDER
CURVE EVEN IF YOU DO YOU WANT STOP.
JUST ONE.
LET ME JUST DO ONE MORE EXAMPLE LIKE THAT.
CURVE Y-EQUALS ONE OVER THE SQUARE ROOT OF X-.
THAT WHAT'S IT LOOK LIKE?
GOES TO INFINITY.
SMALL.
LET ME TAKE THE
NOW WHAT I DRAW
IT, IT HAS THE VERTICAL ASYMPTOTE AM
THAT'S WHAT THE CURVE LOOKS LIKE.
AS X-GETS
SO I COULD ASK, FOR EXAMPLE, LET ME SAY FROM, I COULD
ASK FOR THE AREA UNDER THE CURVE FROM A, SOME LITTLE POSITIVE
NUMBER, UP TO ONE.
THAT'S A PERFECTLY REASONABLE AREA TO ASK
FOR, SO IS THE AREA FROM A-TO ONE, WELL, LET'S DO IT.
FROM A-TO ONE, OF D-X-OVER THE SQUARE ROOT OF X-.
NICE SIMPLE ONE.
INTEGRAL
SO THAT'S A
SO WHO CAN DO THE INTEGRAL OF X-TO THE MINUS
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ONE-HALF, WHAT IS THAT?
IT'S X-TO THE PLUS ONE-HALF TIMES TWO.
SO DIFFERENTIATE THIS AND YOU GET ONE ROVER THE SQUARE ROOT OF
X-.
FROM ONE TO A, SO THAT'S TWO TIMES ONE, MINUS TWO TIMES THE
SQUARE ROOT OF A-.
SO THAT'S A NICE SIMPLE THING.
LET ME ASK
THE QUESTION WHAT HAPPENS TO THE AREA AS A-GOES TO ZERO?
IN
OTHER WORDS I WANT TO FIND THE AREA UNDER THIS CURVE ALL THE WAY
UNDER THE ASYMPTOTE.
EVERYTHING.
EVEN THOUGH IT'S GOING TO
INFINITY, DOES THAT MAKE SENSE TO FIND THAT AREA UNDER THAT?
WELL, THE ANSWER IS AS BEFORE, LET ME TAKE THE LIMIT AS A-GOES TO
ZERO, OF MY FORMULA FOR THE AREA, SO LET ME, SO THE AREA UNDER
ONE OVER THE SQUARE ROOT OF X-FROM ZERO TO ONE IS GOING TO BE THE
LIMIT OF THIS THING, TWO MINUS TWO SQUARE ROOT OF A-.
THE LIMIT OF THAT?
AS A-GOES TO ZERO?
GOES TO ZERO SO IT'S JUST TWO.
AS A-GOES TO ZERO THIS
SO THE AREA UNDER THIS CURVE
GOING ALL THE WAY OVER HERE IS JUST TWO.
REASONABLE THING TO ASK.
SO WHAT'S
IT'S A PERFECTLY
SO SOMETIMES YOU CAN HAVE FINITE AREAS
EVEN WHEN SOMETHING GOES TO INFINITY THAT WAY OR WHEN GOES ALL
THE WAY OVER TO INFINITY THAT WAY.
OKAY.
ANY QUESTIONS ABOUT SECTION 6.4?
SECTION OF THE BOOK.
SO THE LAST
SO WE'VE DONE A BUNCH OF APPLICATION OF
INTEGRATION THROUGHOUT THIS CHAPTER.
MORE.
SO NOW GOING TO DO SOME
HERE'S THE IDEA, IT TURNS OUT THERE ARE A LOT OF THINGS
THAT WE WANT TO ESTIMATE CAN BE WRITTEN IN A WAY THAT YOU
RECOGNIZE, CAN BE WRITTEN AS RIEMANN SUM.
EVEN IF THE QUESTION
DOESN'T SAY ANYTHING ABOUT AREA, IF YOU WRITE IT DOWN IT LOOKS
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LIKE A RIEMANN SUM.
SO WHAT'S A RIEMANN SUM?
THE SOME LITTLE GUY THERE, LITTLE DELTA X-.
IT'S A SUM WHERE
AND THESE X-OF I-S
HAVE SOMEHOW SPACED OUT BETWEEN TWO NUMBERS A-AND B, EVENLY
SPACED BETWEEN A-AND B-.
LOOKS LIKE THIS.
INTEGRAL.
AND YOU CAN RECOGNIZE WHAT WE WANTS
AND THEN, OF COURSE, THAT APPROACHES THE
FROM A-TO B-OF F-OF X-D-X-.
SO EVEN THOUGH THERE
WAS NOTHING, THE PROBLEM DIDN'T HAVE ANYTHING TO DO WITH AREA YOU
CAN STILL THINK OF IT THIS WAY AND SAY I CAN DO A INTEGRAL.
LET'S JUST TRY AN EXAMPLE.
SO
LET'S TALK ABOUT FINDING THE AVERAGE
VALUE OF A FUNCTION.
SO I THINK I HOPE WE ALL KNOW WHAT AN AVERAGE IS.
LET'S
SAY THE AVERAGE OF N-NUMBERS IS JUST SUM Y-ONE PLUS Y-TWO PLUS
DOT DOT DOT DIVIDE (ON BOARD).
SO THE QUESTION IS CAN WE EXTEND
THAT IDEA TO THE AVERAGE OF A FUNCTION.
WE WANT TO DO, EXTEND
THIS IDEA TO THE AVERAGE OF A FUNCTION F-OF X-IS SOME INTERVAL.
AND SOME INTERVAL TO B-.
SO LET ME DO AN EXAMPLE.
LET'S TRY
THE F-OF X, AVERAGE OF F- IN THE INTEGRAL FROM ZERO TO ONE.
LET'S JUST THINK OF WHAT THAT WOULD MEAN, I WOULD TAKE A WHOLE
BUNCH OF VALUES OF X-AT EVENLY SPACED POINTS.
SO TAKE F-AT ONE
OVER N-AND F-AT TWO OVER N-AND F-AT THREE OVER N-ALL THE WAY UP
TO F-OF N-OVER F-WHICH IS ONE.
IN THERE.
TAKE N-DIFFERENT VALUES SPACED
AND I TAKE THE AVERAGE OF ALL OF THEM.
IT'S A
PERFECTLY REASONABLE IDEA OF WHAT IT MEANS TO TAKE THE AVERAGE OF
A FUNCTION.
SO THIS IS THE AVERAGE OF F-AT ALL THESE NICE
EQUALLY SPACED POINTS BETWEEN ZERO AND ONE.
THAT'S A PERFECTLY
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REASONABLE THING.
LARGER?
SO WHAT HAPPENS TO THIS AS N-GETS LARGER AND
THIS IS A AVERAGE FOR A PARTICULAR VALUE OF N-.
RECOGNIZE THAT AS A RIEMANN SUM.
IT'S A RIEMANN SUM.
DO YOU
THAT'S EXACTLY WHAT THAT IS,
IS SO LET ME JUST TAKE THE LIMIT AS N-GOES
TO INFINITY, THIS RIEMANN SUM, THERE WE GO, (ON BOARD).
WHAT'S
THE VALUE OF THAT LIMIT, RECOGNIZING IT'S A RIEMANN SUM?
IT'S
THE AREA UNDER SOME -- IT'S THE INTEGRAL FROM WHAT TO WHAT?
STARTING OFF AT ZERO AND GOING UP TO ONE, INTEGRAL OF F-OF
X-D-X-.
THERE'S THE AVERAGE VALUE, SO THIS I HOPE IS
INTUITIVELY THE AVERAGE VALUE OF F-AT N-POINT EVENLY SPACED
BETWEEN, ALMOST ZERO, ALL THE WAY UP TO ONE.
(ON BOARD).
I'M
ADDING N-VALUES, TAKE THE SUM, THAT'S THE DEFINITION OF AVERAGE,
GETTING CLOSER TO CLOSEER THIS.
(QUESTION).
IF I DRAW WHERE
THESE POINTS ARE ON THE X-AXIS FROM ZERO TO ONE, HERE THEY ARE,
THERE'S ONE OVER N, THERE'S TWO OVER N-DOT DOT DOT AND HERE
N-OVER N-.
RIGHT?
THAT IS WHERE THESE POINTS ARE.
NOW I'M
DOING THE RIEMANN SUM, IF THIS IS FUNCTION OF F, THAT'S A
PERFECTLY (INAUDIBLE) OF THE AVERAGE.
ONE.
IF I'M GOING FROM ZERO TO
SO LET'S ASK WHAT ABOUT THE AVERAGE IF NOW I WANT TO TAKE,
I DON'T WANT TO GO FROM A-TO B-.
(INAUDIBLE).
I WANT TO FIND THE AVERAGE
TAKE THE AVERAGE OF THE VALUE OF THE FUNCTION AT
ALL THOSE POINTS.
TAKE LOTS AND LOTS OF (INAUDIBLE).
SO THE
AVERAGE IS GOING TO BE, LET ME CALL THIS POINT X-ONE, X-TWO,
X-THREE.
ALL THE WAY UP TO X-N-.
EVENLY SPACED.
SO THE
AVERAGE IS GOING TO BE ONE OVER N-TIMES THE SUM OF F-OF X-ONE,
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PLUS F-OF X-N-.
SO THERE'S THE DEFINITION OF THE AVERAGE.
THIS IS A RIEMANN SUM?
ALMOST.
SO
FOR IT TO BE A RIEMANN SUM,
WHAT DO YOU HAVE TO MULTIPLY IT BY?
I HAVE TO MULTIPLY BY HOW
WIDE ONE OF THOSE INTERVALS IS, WHICH WE CALLED DELTA X-BEFORE.
WHAT I WANT TO DO IS WRITE THIS AS DELTA X-TIMES (ON BOARD)
BECAUSE THIS IS THE RIEMANN SUM.
BUT YOU HAVE TO HAVE THIS GUY.
SO THE QUESTION IS, HOW BIG IS DELTA X, WHAT DOES DELTA X-HAVE TO
DO WITH ONE OVER N-.
N-PIECES.
IT'S THE INTEGRAL FROM A-TO B-DIVIDED INTO
SO DELTA X-IS THE LENGTH OF INTEGRAL DIVIDED BY N-.
SO THAT'S NOT QUITE THIS.
SO IF I WRITE THIS NOW AS B-DIVIDED
BY A-OVER N, THIS IS, SORRY, SO THIS IS WHAT I WANT.
WRITE IT OVER HERE.
SO WHAT I WANT IS THIS.
NEED TO WRITE THAT AS RIEMANN SUM.
AND LET ME
(ON BOARD).
THERE IT IS.
I
DELTA X, SO I
CAN WRITE THIS, IF I DIVIDE BOTH SIDES BY B-MINUS A-I GET DELTA
X-OVER B-MINUS A-EVALUATING ONE OVER N-.
SO THERE IT IS.
NOW
IF I TAKE THE LIMIT AS N-GOES TO INFINITE, I GET ONE OVER B-MINUS
A-TIMES THE INTEGRAL FROM A-TO B-OF F-OF X-D-X-.
I GET THIS
EXTRA FACTOR OF ONE OVER THE LENGTH OF THE INTEGRAL.
THE AVERAGE THAT I WANT.
THIS IS
THERE'S A ONE OVER N-FACTOR.
BUT TO
DO THE INTEGRAL, IT'S DELTA X-.
DIVIDE BY B-MINUS A-.
STUDENT:
AND TO GET ONE OVER N-I HAVE TO
DELTA X-OVER B-MINUS A-
IS THAT THE DEFINITION.
PROFESSOR:
SO THIS NOW IS THE AVERAGE OF THE FUNCTION IN THE
INTEGRAL FROM B--- IF B-MINUS A-IS ONE, MY FIRST EXAMPLE THIS
WASN'T THERE, UP HERE, THIS WAS ONE OVER ONE MINUS ZERO.
SO
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DIDN'T SHOW UP.
THIS IS THE GENERAL.
DIVIDE BY THE LENGTH.
THIS IS THE AVERAGE OF THE VALUE OF THE FUNCTION F-EQUALED
(INAUDIBLE).
LET ME DO AN EXAMPLE OF THAT.
SO I'M GOING TO USE THIS FOR A PARTICULAR, I'M GOING TO DO
POPULATION GROWTH SINCE WE'VE USED THAT EXAMPLE MANY TIMES.
1998 WORLD POPULATION WAS 5.9 BILLION PEOPLE.
OKAY.
WAS GROWING AT THAT TIME AT 1.34 PERCENT PER YEAR.
POPULATION IS GROWING.
OKAY.
IN
AND IT
SO
SO THE QUESTION I'M GOING TO ASK
IS WHAT IS THE AVERAGE WORLD POPULATION OVER 30 YEARS STARTING IN
1998.
IN OTHER WORDS FROM 1998 TO 2028.
AVERAGE POPULATION OVER THOSE 30 YEARS.
I WANT TO KNOW THE
THAT'S THE AVERAGE, I
WANT TO WRITE DOWN THE FUNCTION WHICH IS THE WORLD POPULATION.
I NEED P-OF T EQUALS WORLD POPULATION, TIME T AT YEAR T, AND
WE'LL SAY IT'S THE YEAR SINCE 1998.
AND WHAT I WANT IS THE AVERAGE.
T EQUALS ZERO IS 1998.
SO THE AVERAGE IS GOING TO BE
THE AVERAGE OVER 30 YEARS.
SO I'M GOING TO GO FROM ZERO TO
30.
AND INTEGRATE IT.
TAKE THE POPULATION.
SOMETHING OUT FRONT.
OVER HERE.
BUT I HAVE TO PUT
WHAT AM I GOING TO DIVIDE BY?
ONE OVER 30.
SO THERE, ONCE I FIGURE OUT P-OF T, I'LL DO THE
INTEGRAL AND GETS THE AVERAGE POPULATION, NOT THE MAX, THE
AVERAGE.
SO THE AVERAGE IS AN INTERESTING NUMBER BECAUSE IT
TELLS YOU HOW MANY RESOURCES YOU HAVE TO PLAN FOR AND THAT SORT
OF THING.
SHOULD WE FIGURE THIS OUT?
SO IF I KNOW SOMETHING'S
GROWING AT A CERTAIN RATE, THEN I KNOW FROM AN EARLIER CHAPTER
IT'S GROWING EXPONENTIALLY.
IT'S GROWING -- IT'S GOING TO BE
14
SOME MULTIPLE OF THAT.
THIS IS STUFF FROM CHAPTER FIVE.
SO
THIS IS GOING TO BE MY EXPONTENTIAL GROWTH AND THAT COMES FROM
WHAT I TOADS YOU IN THE DATA.
SHOULD I MULTIPLY IT BY?
MULTIPLIED BY SOMETHING.
5.9 BILLION.
WHEN T EQUALS ZERO AT
THE BEGINNING THE POPULATION IS 5.9.
T IT GROWS BY ABOUT 1.34 PERCENT.
WHAT
EVERY YEAR YOU ADD ONE TO
SO THE AVERAGE THEN IS,
LET'S, I HAVE ONE 30TH AND I HAVE .. ZERO (ON BOARD).
NEED ONE MORE FACTOR TO MAKE THIS WORK.
TO MAKE THIS WORK OUT RIGHT?
YOU CAN PLUG IN 30.
BE 7.26 BILLION.
STUDENT:
AND I
WHAT DO YOU NEED TO DO
ONE DIVIDED BY (ON BOARD).
SO NOW
PLUG IN ZERO, SUBTRACT, AND IT COME OUT TO
THAT'S THE NUMBER.
PROFESSOR:
ARE YOU MULTIPLYING BETWEEN THE 5.9 AND ONE -YEAH, THIS IS ALL, JUST PULLING OUT THE FACTOR THERE.
TIMES, TIMES.
OKAY.
THERE'S THE EXAMPLE OF AN STRONG
FUNCTION, FUNCTION WHO'S AVERAGE IS (INAUDIBLE).
QUESTIONS ABOUT AVERAGES?
OKAY ANY
OKAY.
SO THE LAST APPLICATION.
TO DEFINE WHAT I MEAN BY THAT.
SOLIDS OF REVOLUTION.
SO I HAVE
THE IDEA IS YOU TAKE A CURVE
F-OF X-FROM A-TO B, AND YOU ROTATE IT OUT OF THE PLANE TO GET A
SOLID.
I'M GOING TO DRAW PICTURE HERE.
HERE'S THE X-AXIS FROM A-TO B-.
SO
I'M GOING TO TAKE A REALLY
REALLY SIMPLE CURVE, A STRAIGHT LINE.
IS ROTATE THIS OUT OF PLANE.
AND MAKE A CIRCLE.
THERE IS A WORD.
AND WHAT I'M GOING TO DO
BREAK IT OUT LIKE THIS, AND THEN,
WHAT I'M GOING TO GET IS, A CYLINDER.
SO
WHEN I TAKE THIS LINE AND ROTATE IT OUT IT JUST TRACES OUT THE
15
SIDES OF THE CYLINDER.
DO ANOTHER EXAMPLE.
AND I GET A CYLINDER.
SO LET ME JUST
SUPPOSE I TAKE THIS CURVE, ALSO A STRAIGHT
LINE BUT NOW IT GOES FROM ZERO UP TO SOMETHING.
AND I'M GOING
TO TAKE THIS THING AND ROTATE IT OUT OF BOARD.
AND BACK AGAIN.
WHAT SHAPE DO YOU GET NOW?
IT'S GOING TO BE THAT POINTS ALWAYS
GOING TO STAY THERE BUT I'LL GET A CIRCULAR CONE.
SHAPE I GET.
AND ALL THE CROSS SECTIONS ARE CIRCLES.
THE CROSS SECTIONS.
IT CLEAR.
THAT'S THE
SO LET ME DO ANOTHER EXAMPLE JUST TO MAKE
LET'S SUPPOSE I TAKE A SEMICIRCLE.
AND ROTATE IT OUT OF THE PLANE.
GET A SPHERE.
NICE CURVE.
WHAT DO I GET P?
SO WRITE THIS DOWN.
ROTATE IT YOU GET A SPHERE.
GET A CYLINDER.
A SPHERE.
IF YOU TAKE A HORIZONTAL LINE, YOU
AND IF YOU TAKE A SLANTED LINE, YOU GET A CONE.
THERE'S A-AND THERE'S B-.
WHAT DO I GET?
SO
YOU GET A WINE GLASS.
SO YOU CAN GET ANY, SHAPE THAT YOU WANT.
GENERAL IDEA.
I
IF TAKE A SEMICIRCLE AND
AND LET ME JUST DO ONE MORE EXAMPLE TO MAKE IT CLEAR.
OKAY.
ALL THE
SO WHY AM I TELLING YOU ALL THIS?
THAT'S THE
THE QUESTION
IS, I'D LIKE TO FIGURE OUT THE VOLUMES OF ALL THESE SOLIDS.
I
WANT A NICE SIMPLE WAY IT FIGURE OUT WHAT THE VOLUMES ARE SO YOU
KNOW HOW MUCH WINE I CAN PUT IN THE WINE BOTTLE.
A FORMULA FOR THE VOLUMES OF SOLIDS OF REVOLUTION.
SO THE GOAL IS
SO LET ME
JUST, IT'S JUST GOING TO BE A RIEMANN SUM BUT FOR VOLUMES INSTEAD
OF AREAS.
SO LET ME RETRACE THE STEPS FOR AREAS AND THEN I'LL
DO THE SAME STEPS FOR VOLUME.
SOLID REVOLUTION.
HERE'S MY CURVE AND HERE'S ANY
SO LET ME REMIND OURSELF THE STEPS TO GET THE
16
AREA UNDER OF CURVE.
I IMAGINE ALL OF THESE LITTLE RECTANGLES
TO GET MY RIEMANN SUM.
IF I TAKE ANY PARTICULAR RECTANGLE, IT
HAS A WIDTH, DELTA X, AND IT HAS A HEIGHT, F-OF X-OF I-.
IF
THIS IS, THAT POINT THERE IN THE MIDDLE IS X-OF I-THAT PARTICULAR
LITTLE RECTANGLE HAS A AREA WHICH IS THE HEIGHT TIMES THE BASE.
SO THE AREA OF ONE RECTANGLE IS JUST DELTA X-F-OF X-OF I-.
WHOLE AREA, SO THAT'S ONE AREA.
IS JUST ADD THEM ALL UP.
AND THE WHOLE AREA, OF COURSE,
THIS IS JUST REVIEW.
BEHOLD I GET THIS INTEGRAL.
THE
AND LOW AND
THAT'S WHAT WE DID FOR AREAS.
WANT GOING TO DO THE SAME THING FOR VOLUMES.
I
I JUST HAVE TO
FIGURE OUT HOW IT BREAK IT UP INTO LITTLE THING EACH OF WHOSE
VOLUMES I UNDERSTAND.
SO HERE'S THE IDEA.
LOTS OF LITTLE SLICES THAT LOOK LIKE THIS.
THIN CYLINDRICAL SLICES.
BREAK IT UP INTO
(ON BOARD).
WHAT I'M GOING TO DO IS TAKE ANY ONE
OF THESE, IT'S GOING TO LOOK LIKE A LITTLE FLAT DISK.
WANT TO DO NOW, AND IT HAS A RADIUS.
RADIUS IS F-OF X-OF I-.
LITTLE
WHAT I
WHAT'S THE RADIUS?
THE
THAT POINTS RIGHT THERE IS X-OF I-.
AND ITS WIDTH IS DELTA X-.
LITTLE CIRCULAR SLAB.
IF I TAKE
ONE OF THOSE, JUST ONE OF THESE LITTLE CIRCULAR SLABS, WHAT'S THE
VOLUME?
WHAT'S THE VOLUME OF SOMETHING THAT'S CIRCULAR THAT WAY,
AND I KNOW THE RADIUS AND I KNOW THE HEIGHT.
WHO REMEMBERS?
IT'S THE AREA OF THE BASE, TIMES THE HEIGHT, THICKNESS.
GOING TO BE DELTA X-TIMES WHAT?
AND THE BASE?
SO IT'S
WHAT'S THE AREA OF THE CIRCLE
IT'S PI TIMES F-OF X-OF I-SQUARED.
THE AREA OF THIS CIRCLE, THE BASE.
THAT'S JUST
TIME THE THICKNESS, DELTA
17
X-.
SO THE WHOLE AREA, NOW IF I ADD UP ALL THOSE, I'M GOING TO
DO A SUM.
AND IT'S GOING TO BE F-OF X-ONE SQUARED PLUS F-OF
X-TWO SQUARED PLUS F-OF X-OF N-SQUARED.
ADD THEM ALL UP.
SO NOW WHAT HAPPENS IS I LET N-GOES TO INFINITY?
A RIEMANN SUM, RIGHT?
THIS LOOKS LIKE
IT'S DELTA X, THERE'S A PI THAT FACTOR
OUT, DELTA X-TIMES SUM OF SOME FUNCTION EVALUATIONS.
PULL OUT THE PI.
SO LET ME
IT'S GOING TO BE A INTEGRAL FROM A-TO B-.
SHOULD HAVE SAID FROM A-TO B-.
FROM A-TO B-.
FUNCTION WHOSE VALUES I'M ADDING UP?
IS.
AND
BUT WHAT'S THE
F-OF X-SQUARED.
THERE'S THE ANSWER FOR THE VOLUME.
I
THERE IT
A NICE SIMPLE FORMULA.
JUST INTEGRATE F-SQUARED INSTEAD OF F-AND MULTIPLY BY PI.
STUDENT:
WHAT ABOUT DELTA X-
PROFESSOR:
TO INFINITY.
THAT HAS PART THE RIEMANN SUM.
REMEMBER WE LET N-GO
DELTA GETS SMALLER AND SMALLER AND THE WHOLE THING
CONVERGES TO THE INTEGRAL.
NOW VOLUMES ARE EASY TOO.
THE SAME THING HAPPEN OVER HERE.
LET'S DO SOME.
SO I'M GOING TO USE
THESE EXAMPLES.
SO LET ME MAKE THIS GO FROM ZERO TO H-.
BE THE LENGTH, THE HEIGHT OF THE CYLINDERS.
DISTANCE BE R, THE RADIUS.
SO H-IS GOING TO
AND LET ME LET THIS
SO WHAT'S MY FUNCTION FOR THIS ONE?
F-OF X-EQUALS, SO THE QUESTION IS WHAT'S THE NAME OF THIS
FUNCTION, JUST THE HORIZONTAL LINE AT R, F-OF X-EQUALS R-.
SO
THE VOLUME OF THE CYLINDER OF RADIUS R-IN HEIGHT H-IS GOING TO
BE, LET ME USE THE FORMULA, THERE'S A PI, WITH INTEGRAL FROM ZERO
TO H, OF THIS FUNCTION F-OF X-EQUALS R-SQUARED.
R-IS JUST
18
CONSTANT.
R-SQUARED D-X-.
SO WHO CAN DO THAT?
WHAT'S THE
INTEGRAL OF A, THIS IS PI R-SQUARED TIME THE INTEGRAL OF ZERO TO
H-OF ONE D-X-.
JUST X-.
I HOPE?
AND WHAT'S THE INTEGRAL OF ONE?
AND THAT'S PIE R-SQUARED H-.
(ON BOARD) IT'S
DOES THAT LOOK FAMILIAR
IT'S JUST THE AREA OF PI R-SQUARED IS THE AREA THE
CIRCULAR BASE TIMES THE HEIGHT OF CYLINDER.
WE GET THE FORMULA
THAT'S PROBABLY, YOU'RE FAMILIAR WITH FROM GEOMETRY.
A CONE.
LET ME DO
YOU MAY OR MAY NOT KNOW THE FORMULA FOR THIS.
SO
AGAIN, LET'S SUPPOSE THAT THE HEIGHT OF THE CONE IS H-.
SO
INTEGRATE FROM ZERO TO H-.
OF CONE IS R-.
AND LET'S SUPPOSE THE RADIUS OF BASE
SO IT'S GOING TO BE A CONE OF HEIGHT H-AND
RADIUS R-AT THE BASE.
I WANT TO FIND ITS VOLUME.
NEED, THIS DISTANCE HERE IS GOING TO BE R-.
FUNCTION THAT GIVES ME THIS STRAIGHT LINE.
TO DO THAT I
WHAT IS THE
WHO CAN TELL ME WHAT
STRAIGHT LINE GOES THROUGH THE ORIGIN AND THROUGH POINT R-.
THIS POINT HERE IS R-COMMA H-.
H-COMMA R, SORRY.
GOES OUT
SO
H-UP R-.
LINE?
WHAT, SO WHAT'S THE FUNCTION GIVES ME THAT STRAIGHT
EQUATION OF A STRAIGHT LINE THERE?
SLOPE, SO WHAT'S THE SLOPE OF THIS LINE?
IT'S X-TIMES THE
R-OVER H-.
SO I HAVE,
THERE'S THE FUNCTION THAT GIVES ME, THE BOUNDARY OF ANY CIRCULAR
CONE.
SO LET'S DO THE VOLUME OF A CONE.
AND AGAIN IT'S GOING
TO BE RADIUS R-HEIGHT H, SO I GET PI TIMES THE INTEGRAL FROM ZERO
TO H-OF WHATEVER THAT WAS OVER THERE, R-OVER H-TIMES X-QUANTITY
SQUARED D-X-.
(ON BOARD).
H-S ARE CONSTANT.
SEE WHAT WE GET.
FACTOR THOSE OUT FRONTS.
SO THESE R-AND
R-SQUARED OVER
19
H-SQUARED.
I'M LEFT WITH X-SQUARED.
X-SQUARED?
IT'S X-CUBED OVER THREE, EVALUATED AT H-AND ZERO.
SO I PLUG IN H-.
ZERO.
I GET H-CUBED OVER THREE, PLUG IN ZERO I GET
SO I GET PI R-SQUARED H-SQUARED H-CUBED OVER THREE.
SOMETHING CANCEL HERE, RIGHT?
H-.
SO WHAT'S THE INTEGRAL OF
SO I GET PI OVER THREE R-SQUARED
THERE IT IS.
STUDENT:
PROFESSOR:
WHERE DO WE GET R-OVER H-XSO R-OVER H-IS THE EQUATION OF THIS LINE.
SLOPE OF THE LINE WHICH IS THE BOUNDARY OF THE CONE.
THE CONE GOES OUT TO H-.
WHERE IT COMES FROM.
R-SQUARED H-.
AND IT GOES UP TO R-.
SO DID I GET AT THAT RIGHT?
IT IS THE
BECAUSE
SO THAT'S
PI OVER THREE
LET US DO, LET'S DO THE VOLUME OF THE SPHERE.
SO HERE'S IS MY SPHERE OF RADIUS R-.
SO WHAT I WANT IS A
FUNCTION FOR EQUATION FOR THE VERTICAL, FOR THE BOUNDARY.
WRITE DOWN X-SQUARED PLUS Y-SQUARED EQUALS R-SQUARED.
SO
I KNOW
THAT'S HOW YOU DEFINING A SEMICIRCLE AM IF I SOLVE FOR Y-I GET
Y-IS THE SQUARE ROOT (ON BOARD).
BOARD).
SO THIS IS F-OF X-.
(ON
SO THAT TELLS ME I SHOULD COME HERE AND DO THE INTEGRAL
OF PI, SO WHAT ARE THE LIMIT OF INTEGRATION?
DO I INTEGRATE FROM?
FOR THE SPHERE WHAT
I WANT TO INTEGRATE FROM ALL THE WAY OVER
HERE TO ALL THE WAY OVER THERE WHICH IS FROM WHERE TO WHERE?
FROM ZERO.
R-.
FROM MINUS R-TO R-.
NOT
SO INTEGRATE FROM MINUS R-TO
AND NOW MY FUNCTION IS THE SQUARE ROOT OF R-SQUARED MINUS
X-SQUARED.
FORTUNATELY I'M GOING TO SQUARE THAT SQUARE ROOT
WHICH WILL MAKE THE INTEGRAL EASIER TO DO.
PI TIMES INTEGRAL
20
FROM MINUS R-TO R-OF R-SQUARED MINUS X-SQUARED D-X-.
DO IT.
LET'S JUST
THE INTEGRAL OF R-SQUARED IS R-SQUARED X-.
AND
INTEGRAL OF MINUS X-SQUARED IS MINUS X-CUBED OVER THREE.
EVALUATED AT M, EVALUATED AT MINUS R-AND SUBTRACT.
I
WHEN ALL IS
SAID AND DONE WHAT YOU THINK THE ANSWER IS?
ANYBODY?
VOLUME OF SPHERE?
THAT'S WHERE IT
COMES FROM.
FOUR THIRDS PI R-CUBED.
WHAT'S THE
IF YOU EVER FORGET THE VOLUME OF A SPHERE YOU CAN
DO IT THIS WAY BUT YOU PROBABLY DO REMEMBER IT'S FOUR THIRDS PI
R-CUBED.
ONE MORE EXAMPLE LIKE THIS.
I'M GOING TO TAKE AN ELLIPSE.
SQUASHED CIRCLE.
SO IT'S GOING TO GO OUT TO A-IN THAT C AND
B-IN THAT DIRECTION.
CALL AN ELLIPSOID.
AND SO IT'S GOING TO BE A
ROTATE IT AND I'M GOING TO GET SOMETHING
AND I WANT FIND THE VOLUME OF THAT POTATOES
THING. SO WHAT IS THE EQUATION FOR AN ELLIPSOID IF A-AND B-WERE
EQUAL THIS WOULD BE A CIRCLE OF RADIUS A-.
NOW WHEN X-IS A-Y-IS
ZERO.
IT GOES THROUGH THAT POINT.
WHEN X-IS ZERO -- THIS IS
HOW -- SO LET ME JUST, I NEED TO FIND AN EQUATION FOR THAT TOP
LINE.
LET ME SOLVE.
MULTIPLY, TAKE THE SQUARE ROOT.
THAT Y-IS THE SQUARE ROOT OF B-SQUARED (ON BOARD).
I GET
AND THAT'S
MY FUNCTION F-OF X-THAT I GOT TO PLUG INTO FORMULA FOR VOLUME.
SO THE VOLUME OF THE ELLIPSOID IS GOING TO BE PI.
THE LIMIT OF INTEGRATION FROM WHERE TO WHERE?
F-OF X SQUARED.
AND WHAT ARE
NEGATIVE A-TO A-OF
SO IT'S AGAIN I'M GOING TO GET A SQUARE ROOT
BUT I GET TO SQUARE IT RIGHT AWAY AND MAKE IT GO AWAY.
CANCELS.
(ON BOARD).
SO THAT
NOW I'LL INTEGRATE TERM BY TERM, WHAT IS
21
THE INTEGRAL OF B-SQUARED, B-SQUARED TIMES X-.
INTEGRAL OF X-SQUARED?
AND WHAT IS THE
IT'S X-CUBED OVER THREE.
AND I'LL
EVALUATE THAT AT PLUS A-AND MINUS A, SUBTRACT AND WHEN ALL IS
SAID AND DONE, I GET FOUR THIRDS PI B-SQUARED A-.
SIMPLE FORMULA.
CHECK.
SO A NICE
SO IF I COME BACK HERE, LET'S DO A SANITY
IF A-EQUALS B, WHAT DOES THIS ELLIPSOID?
IT'S A CIRCLE.
SO I OUGHT TO GET A SPHERE WHEN I ROTATE IT, THE SPHERE THE
RADIUS A-.
CUBED.
IF B-EQUALS A-I GET FOUR THIRDS PI TIMES RADIUS
I GET THE SAME THING I GOT OVER THERE. YOU SHOULD, THAT
GIVES YOU A NICE WARM FEELING YOU DID THE MATH RIGHT.
DOESN'T CHANGE VERY MUCH TO GET THIS VOLUME.
END OF THE COURSE EXCEPT FOR REVIEW.
STUDENT:
OKAY.
SO IT
THAT'S THE
ANY QUESTIONS?
PROFESSOR:
DO YOU HAVE OFFICE HOURS TOMORROW.
YES.
I APOLOGIZE I HAVE A SICK CHILD AT HOME ON
MONDAY.
I'LL BE BACK.
SO WE HAVE A CHOICE NOW.
CHOICE.
I CAN DO SAMPLE PROBLEMS FROM THE FINAL.
A THREE WAY
TAKE THE
PLEASURE AWAY FROM YOU OF DOING SAMPLE PROBLEMS FOR THE SAMPLE
FINAL OR I COULD DO MY REVIEW OF WHAT I THINK ARE THE MOST
IMPORTANT POINTS SINCE THE LAST MIDTERM OR WE COULD GO HOME.
REVIEW.
OKAY.
I TAKE THAT AS A SILENCE IS CONSENT.
STUDENT:
SHOULD WE ALWAYS KNOW EQUATIONS FOR THE FINAL.
LIKE
THE EQUATION.
PROFESSOR:
THEM.
I'M DOING THESE AS A EXAMPLES SO YOU KNOW HOW TO DO
AND FOR THE FINAL YOU SHOULD KNOW HOW TO DO THEM.
IF I
GIVE YOU A SURFACE OF REVOLUTION AND I GIVE YOU F-OF X-I MAY ASK
22
WHAT'S THE VOLUME.
BUT I WON'T, ASK YOU TO DO A, WHAT'S THE
VOLUME OF A ELLIPSOID.
I WON'T SPELL IT OUT FOR YOU.
ANY
OTHER QUESTIONS?
OKAY, SO THIS IS, GIVEN THE AMOUNT OF TIME I HAVE LEFT, I
WON'T COVER EVERY POINT.
LOGARITHMS.
YOU HAVE TO KNOW ABOUT NATURAL
BASIC PROPERTIES AND HOW TO DIFFERENTIATE.
WE
SHOULD ALL KNOW, FOR EXAMPLE, THAT THE NATURAL LOG OF X-PLUS
Y-EQUALS NATURAL LOG OF X-PLUG NATURAL LOG OF Y-.
AND IF YOU DO
THE DIVISION IT'S THE NATURAL LOG OF X-MINUS NATURAL LOG OF Y,
LOGARITHMS.
IF YOU HAVE A EXPONENT YOU CAN PULL IT OUT FRONT.
SO THAT WILL GET YOU MOST OF THE WAY.
DIFFERENTIATE IT TOO.
YOU BETTER KNOW HOW TO
WHICH IS PRETTY EASY, ONE OVER X-.
AND
SO FOR EACH RULE LIKE THIS, OF COURSE, GIVES YOU A RULE ABOUT
INTEGRATION TOO.
SO THAT GOES WITHOUT SAYING.
OKAY.
THE
NEXT MAIN THING TO THINK ABOUT IS APPLICATIONS AND WITH
EXPONENTIAL GROWTH AND DECAY, WE HAD A LOT OF APPLICATIONS, FOR
EXAMPLE, POPULATION GROWTH, RADIOACTIVE DECAY.
COMPOUND
INTEREST, THERE WERE LOTS OF EXAMPLES WHERE THESE MODELS APPLIED.
AND YOU MAY GET SUCH AN EXAMPLE TO WORK OUT.
BUT LET ME BE MORE
SPECIFIC HERE.
STUDENT:
PROFESSOR:
DO WE ONLY HAVE TO KNOW.
IF YOU HAVE, WHAT SHOULD KNOW IS THAT IF YOU HAVE
INTEREST COMPOUNDED AT FIX PERIOD LIKE MONTHS YOU CAN STILL WRITE
THAT AS AN EXPONENT FUNCTION.
IT'S AN EXPONENTIAL FUNCTION.
IT IT'S COMPOUNDED CONTINUALLY
THEY'RE ALL EXPONENTIAL
23
FUNCTIONS.
THAT'S THE MAIN FACTOR.
SO THE MAIN IDEA HERE IS
THAT IF YOU FOR SOME REASON KNOW THAT THE RATE OF GROWTH OF YOUR
FUNCTION IS PROPORTIONAL TO THE VALUE OF THE FUNCTION, AND YOU
ALSO HAPPEN TO KNOW THE VALUE OF FUNCTION WHEN EVERYTHING GETS
STARTED AT ZERO, THEN YOU HAVE A EXPLICIT FORMULA FOR THE
FUNCTION.
SO FIGURING OUT THIS IS THE RATE OF GROWTH IF IT'S
POSITIVE OR THE RATE OF DECAY IF IT'S NEGATIVE, THERE'S THE
INITIAL DATA, YOU SOMEHOW HAVE TO FIGURE THAT OUT FROM THE
STATEMENT OF YOUR PROBLEM.
ANSWER.
AND THEN YOU CAN WRITE DOWN THE
THAT COVERS ALL THESE DIFFERENT THINGS.
SO THE GROWTH, THERE COULD BE DECAY.
AND IF THERE'S DECAY
THERE'S ANOTHER WAY TO THINK ABOUT IT, SOMETHING WE CALL HALF
LIFE.
SO THIS IS THE TIME WHERE THE AMOUNT YOU HAVE IS EXACTLY
HALF OF WHAT YOU STARTED WITH.
DISAPPEARS.
THE HALF LIFE IS WHEN HALF OF IT
IN (ON BOARD).
SO AND THE WAY WE FIGURE THAT OUT IS JUST PLUG THIS
SO JUST SOLVE FOR THAT YOU GET THAT IMPLIES THAT
T ONE-HALF IS JUST MINUS NATURAL LOG OF TWO DIVIDED BY K-.
BOARD).
DECAY.
(ON
IT'S JUST ANOTHER WAY OF TALKING ABOUT HOW FAST THINGS
YOU CAN EITHER TALK ABOUT THE THING UP IN THE
EXPONENTIAL OR THE HALF LIFE.
THEIR RELATION IS VERY SIMPLE.
SO THE OTHER FACT THAT FROM THAT SECTION, THAT WE USED AND YOU
SHOULD REMEMBER, IS THIS LIMIT, WHO CAN REMIND ME OF THE LIMIT AS
H-GOES TO ZERO OF WHAT THAT THING IS.
IT WAS E-.
SO THAT IS
SOMETHING THAT CAN BE PROVEN WITH LOGARITHM AND IT HELPS US DO A
LOT OF LIMIT.
WHERE WE GOT ALL THE FORMULA FOR COMPOUND
24
INTEREST BASED ON THAT THING.
SO THE THIRD IMPORTANT TOPIC IS APPROACHING LIMITS.
DIFFERENT MODEL FOR HOW DO YOU APPROACH LIMITS.
IS CERTAINLY ONE WAY.
EXPONENTIALLY
AND WE ALSO LOOKED AT A VARIATION ON IT
CALLED THE LOGISTIC CURVE.
THING.
WITH
SO THESE WERE MODEL FOR DIFFERENT
LET ME DRAW THE PICTURES.
IF YOU HAVE A CURVE THAT
STARTS AT ZERO AND APPROACHES A LIMIT BUT NEVER QUITE GETS THERE,
THERE ARE TWO WAYS TO TALK ABOUT THAT. ONE WAY AND LET THIS BE Y,
IS TO SAY Y-OF T EQUALS K-TIMES M MINUS Y-OF T.
ZERO.
STARTING AT
AND SO K-TELLS YOU HOW FAST IT'S APPROACHING THE LIMIT.
THE FACT THAT THERE IS M MINUS Y-TELLS YOU M IS THE LIMIT.
YOU'LL NEVER GETS PAST IT.
FLATTER AND FLATTER.
WHEN Y-GETS CLOSE TO -- IT GETS
THE DERIVATIVE, IF YOU CAN FIGURE OUT FROM
THE STATEMENT OF YOUR PROBLEM WHAT THE LIMIT IS AND HOW FAST IT'S
GETTING THERE YOU CAN WRITE DOWN A CLOSED FORM FORMULA FOR THE
SOLUTION.
(ON BOARD).
THERE IT IS.
THAT TELLS YOU, AS
K-GETS BIGGER THAT GETS SMALLER AND THE WHOLE THING GETS CLOSER
TO M.
THAT WAS ONE MODEL WE USED OVER AND OVER AGAIN.
OTHER ONE THAT WE USED ALSO HAD A FINAL LIMIT VALUE M.
COULD START ANYWHERE.
THE
BUT IT
FOR A LITTLE WHILE IT GREW EXPONENTIALLY
AND THEN TURNED OVER AND APPROACHED THE LIMIT SO IT LOOKS LIKE AN
S-CURVE.
WE CALL IT A LOGISTIC CURVE.
THE WAY THIS CURVE
AROSE IS YOU ALSO COULD FIGURE OUT FROM THE STATEMENT OF PROBLEM,
THAT THERE WAS SOME FORMULA FOR ITS DERIVATIVE.
IT IS.
THERE IS THE RATE.
SO HERE, HERE
WHICH SAYS HOW FAST YOU'RE GOING.
25
BUT THERE'S, BUT THEY WERE TWO TERMS HERE.
SO WHEN Y-IS CLOSE
TO M THIS GETS CLOSE TO ZERO AND FLATTENS OUT.
BUT WHEN Y-IS
SMALL, THIS IS LIKE A CONSTANT M, AND IT'S, AND THE GROWTH
PORTION OF Y, SO IT'S GROWING EXPONENTIALLY FOR A LITTLE WHILE
AND THEN SLOWS DOWN AND DEPENDING ON THE WORD PROBLEM YOU CAN
FIGURE OUT WHAT THE LIMIT IS AND HOW FAST IT'S GOING THERE.
YOU CAN, THAT ALSO HAS TO BE GIVEN.
AND
AND ONCE YOU KNOW THAT,
THERE'S ALSO A CLOSED FORM SOLUTION FOR WHAT THIS CURVE LOOKS
LIKE, THAT S-CURVE, IT'S THE LIMIT OF ONE MINUS B-TIMES E-TO THE
MINUS M K-T.
(ON BOARD).
SO AS T GETS BIGGER AND BIGGER, THIS
EXPONENTIAL GETS SMALLER AND SMALLER AND APPROACHES ONE.
WHERE
B-IS A CONSTANT, IT DEPENDS ON YOUR STARTING POINT AND IT DEPENDS
ON M.
AND THERE'S SOME ALGEBRA FOR FIGURING IT OUT BUT I WON'T
REPEAT IT.
STUDENT:
PROFESSOR:
IS IT ONE PLUS B-OR ONE MINUS BSORRY, ONE PLUS.
THOSE WERE, AND WE DID A BUNCH OF
DIFFERENT EXAMPLES WHERE GIVEN SOME PROBLEM, SOME MODEL OF
POPULATION GROWTH OR INFORMATION BEING EXCHANGED.
YOU CAN
FIGURE OUT M AND FIGURE OUT K-AND THIS WAS THE ANSWER.
TALKED ABOUT ANTI DIFFERENTIATION.
FOR EVERY RULE OF
DIFFERENTIATION THERE WAS ONE THAT WORKED BACKWARDS.
KNOW ALL OF THEM, THESE, BY HEART.
YOU SHOULD
SO THIS IS, DIFFERENTIATE
THIS, THE CONSTANT GOES AWAY, YOU GET BACK TO THAT.
NOT WORK WHEN ARTICLE IS MINUS ONE.
THEN WE
THIS DOES
YOU GET A A LOGARITHM IN
THAT CASE WHEN THE EXPONENTS IS MINUS ONE.
AND E-TO THE K-X, SO
26
THOSE ARE THE BASIC RULES OF ANTI DIFFERENTIATION PLUS SUM AND
DIFFERENCES, ALL THOSE KINDS OF RULES.
OR MINUS G-OF X-.
SO INTEGRAL, F-OF X-PLUS
D-X-EQUALS INTEGRAL OF F-PLUS OR MINUS G-OF
THESE SORTS OF THINGS.
(ON BOARD).
FACTORING OUT.
THOSE
SHOULD ALL BE PRETTY FAMILIAR.
AND I SHOULD SAY THAT, I ALSO SKIPPED THIS, THIS IS THE
DEFINITION, IF SOMEBODY HANDS YOU LITTLE F, AND YOU FIND BIG F,
THAT'S WHAT DEFINES AN ANTI DERIVATIVE.
OKAY.
THAT'S JUST,
YOU SHOULD KNOW THAT BASIC DEFINITION.
SO AFTER ANTI DIFFERENTIATION, INTEGRATION IS ALMOST A
SYNONYM.
AND RECOGNIZING A RIEMANN SUM AND WHAT IT'S GOOD FOR,
AND THE FACT THEY CAN BE USED TO REPRESENT AREAS, AND FUNDAMENTAL
THEOREM OF CALCULUS.
THOSE ARE THE BASIC IDEAS THAT COME NEXT.
AND SO JUST TO DRAW THE PICTURE I'VE DRAWN IT MANY TIMES TODAY.
(ON BOARD).
SO THE AREA IS VERY CLOSELY APPROXIMATED BY THE
AVERAGE OF ALL OF THOSE FUNCTION VALUES OF ALL THE AREAS OF ALL
THOSE RECTANGLES HERE.
X-IS ONE LITTLE RECTANGLE.
GOES FROM THE INTEGRAL FROM A-TO B-OF F-OF X-D-X-.
THE DEFINITE INTEGRAL MEANS.
AND THAT
THAT'S WHAT
IF YOU KNOW THE ANTI DERIVATIVE,
CAPITAL F-YOU JUST GET IT BY DOING F-OF, CAPITAL F-OF B-TIMES
CAPITAL F-OF A-.
AND THIS IS, IN FACT, THE, THIS EQUALITY HERE
IS FUNDAMENTAL THEOREM OF CALCULUS.
YOU'VE BEEN USING OVER AND
OVER AGAIN WITHOUT SAYING IT, BUT THAT'S WHAT IT IS.
STUDENT:
PROFESSOR:
IF THE M FUNCTION WHAT'S UNDERNEATH THE SUM.
THIS IS THE SUM FROM I-EQUALS ONE UP TO N-.
AND
27
THIS NOTATION HERE IS THE SYNONYM FOR F-OF X-ONE PLUS F-OF X-TWO
PLUS DOT DOT DOT PLUS F-OF X-N-.
FOR THAT THING THERE.
IT'S JUST SHORTHAND NOTATION
AND ANOTHER WAY TO STATE THE FUNDAMENTAL
THEORY OF CALCULUS IS TO SAY IF YOU HAVE A FUNCTION AND FIND IT'S
INTEGRAL AND THEN DIFFERENTIATE IT, YOU GET IT BACK AGAIN.
THESE TWO OPERATIONS CANCEL ONE ANOTHER OUT.
SO THAT'S ANOTHER
WAY TO SAY WHAT THE FUNDAMENTAL THEOREM OF CALCULUS IS.
DONE HERE.
SO
YOU SHOULD BE ABLE TO DO AREA PROBLEMS.
ALMOST
WE'VE DONE
A BUNCH OF THOSE ALREADY.
SO IF THIS FUNCTION IS F-OF X, AND
THIS FUNCTION IS G OF X-.
AND YOU WANT TO FIND THE AREA IN
BETWEEN A-AND B, THEN (ON BOARD) , OF COURSE, THERE ARE
VARIATIONS ON THAT, THAT THEY CROSS, SOMETHING LIKE THAT.
SO
AND VARIATIONS ON FINDING AREAS.
THE LAST TOPIC IS WHAT WE COVERED IN THE LAST TWO OR THREE
LECTURES.
ARE ALL APPLICATIONS OF INTEGRATION.
FOR INSTANCE,
HOW DO YOU GET POSITION IF YOU KNOW THE VELOCITY OF AN OBJECT AND
HOW DO YOU GET THE VELOCITY IF YOU KNOW THE ACCELERATION.
YOU KEEP INTEGRATING.
AND THE AVERAGE VALUE OF THE FUNCTION.
AND WHAT WE JUST DID EARLIER, VOLUMES OF SOLIDS.
OKAY. LOTS OF DIFFERENT EXAMPLES OF THAT.
GOOD PLACE TO STOP.
SO
OF REVOLUTION.
AND I THINK THAT'S A
AND WISH YOU GOOD LUCK ON THE FINAL.
AND
I'M GOING TO POST ON THE WEB PAGE, WE WILL HAVE REVIEW SECTIONS.
AND SOME OF THE TA HAVE ALREADY POSTED THAT BUT WE WILL POST
MORE.
SO HAVE A GOOD CHRISTMAS.
(APPLAUSE).