PROFESSOR: WELCOME BACK TO THE LAST LECTURE OF THE SEMESTER. PLANNING TO DO TODAY WAS FINISH THE BOOK. FINISH SECTION 6.5

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1 1 MATH 16A LECTURE. DECEMBER 9, PROFESSOR: WELCOME BACK TO THE LAST LECTURE OF THE SEMESTER. I HOPE YOU ALL WILL MISS IT AS MUCH AS I DO. SO WHAT I WAS PLANNING TO DO TODAY WAS FINISH THE BOOK. FINISH SECTION 6.5 AND 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. SO LET'S FINISH. SECTION 2.4, ABOUT AREAS. 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. (ON BOARD). SO FROM MINUS ONE TO TWO. SO THIS IS SORT OF SIMPLEST KINDS OF EXAMPLE. IF I DRAW THOSE TWO, Y-EQUALS X-SQUARED MINUS TWO X-. GIVES ME A PARABOLA. MINUS E-TO THE X, GIVES ME THIS CURVE GOING DOWN HERE. THEY NEVER CROSS. I WANT THE AREA BETWEEN MINUS ONE AND TWO. 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,

2 SUBTRACT THAT POINT -- SO THE AREA IS GOING TO BE THE INTEGRAL OF 2 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. TO THE BOTTOM CURVE. THAT'S JUST THE DISTANCE FROM THE TOP 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. AND THAT MINUS MINUS JUST TURNS INTO PLUS. SO IT'S PLUS E-TO THE X-AND INTEGRAL OF THAT IS PLUS E-TO THE X-. SO JUST TERM BY TERM INTEGRATION. SO I PLUG IN TWO. 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. TOP FROM THE BOTTOM AND INTEGRATE. THEN YOU JUST SUBTRACT THE 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. SO LET ME DRAW THAT PARABOLA. IT LOOKS SOMETHING LIKE THIS. SO WHAT I

3 SAY WHAT IS THE AREA BOUNDED BY THOSE TWO CURVES, GEOMETRICALLY 3 IS THIS AREA IN HERE. THAT'S THE ONLY AREA BOUNDED BY THOSE CURVE. THAT GOES ON FOREVER, THIS IS WHAT IS BOUNDS BY THOSE TWO CURVES, THE AREA IN BETWEEN. SO WHAT I HAVE TO DO IS FIGURE OUT WHAT A-IS. AND WHAT B-IS. AND THEN DO THE INTEGRAL FROM A-TO B-. BUT I'M NOT TELLING YOU WHAT A-AND B-ARE U-YOU HAVE TO FIGURE THEM OUT FOUR YOURSELF. LINE INTERSECT THAT PARABOLA. YOU SAY WHERE DOES THIS STRAIGHT 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. SO THAT'S THE SAME PATTERN AS BEFORE. 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. YOU HAVE TO SOLVE QUADRATIC EQUATION TO DO THIS. 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

4 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 4 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. HOW ABOUT FIND THE AREA BOUNDED BY, TWO OTHER CURVES. ANOTHER PARABOLA. AND Y-EQUALS X-CUBED MINUS THREE X-. SO LET ME DOT SAME THING AS BEFORE. I'M NOT TELLING YOU WHAT A-AND B-ARE. LET ME DRAW THE TWO PICTURE. 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. THERE ARE TWO SORTS OF REGIONS THAT MEET AT THAT POINT. I WANT TO FIGURE THAT OUT. THOSE TWO REGIONS. I WANT TO FIGURE OUT THE AREA OF UNION OF THAT'S WHAT THE PROBLEM IS REALLY ASKING FOR. 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. THE TOP CURVE IS X-CUBED MINUS THREE X-. THE BOTTOM IS TWO X-SQUARED. SO LET ME CALL THIS A-AND THAT B-. SO THE AREA I'M GOING TO HAVE TWO AREAS, THE AREA OF A-PLUS AREA OF B-FOR THESE TWO SEPARATE REGIONS. SO

5 WHAT IS THE AREA OF A-GOING TO BE? IT'S GOING TO BE INTEGRAL FROM A-TO ZERO. I HAVE FIGURE OUT LITTLE A-. WHAT'S THE 5 DISTANCE FROM THE TOP TO THE BOTTOM? THE ONE ON TOP IS X-CUBED MINUS THREE X-. THE ONE ON THE BOTTOM IS TWO X-SQUARED. FINE. 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-. AND NOW WHAT'S THE DISTANCE FROM THE TOP TO THE BOTTOM? THE ONE ON THE TOP THEY'VE CHANGED PLACES. NOW TWO X-SQUARED IS ON TOP. SO I HAVE TO DO IT THIS WAY. 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-. YOU HAVE TO SETUP THAT WAY SO ADDING TWO POSITIVE THINGS. 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. THANK YOU. SO I NEED TO SOLVE THIS. LET ME WRITE THIS WAY. ZERO EQUALS X-CUBED MINUS TWO X-SQUARED MINUS THREE X-. 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. AND NOW I CAN SEE IT'S ZERO AT ZERO, THAT'S THIS POINT. WHEN X-EQUALS ZERO THE CURVES CROSS. AND THIS IS NOW A QUADRATIC AND I CAN SOLVE THAT,

6 I GET X-MINUS ONE X, SORRY, PLUS ONE, MINUS THREE. (ON BOARD). THAT'S RIGHT. SO THAT TELLS ME THIS POINTS HERE IS MINUS ONE, AND THAT POINTS IS THREE. SO WHEN I'M FINALLY DONE I PLUG IN 6 A-EQUALS MINUS ONE. AND LITTLE B-EQUALING PLUS THREE, NOW YOU CAN DO THOSE TWO INTEGRALS. WHETHER IT'S ALL SAID AND DONE. I WILL LET YOU DO THE END. AT THIS POINT IT'S SET UP. 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: FOR SKETCHING, WHOLE TAKE THE FIRST DERIVATIVE. PROFESSOR: 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. SO YOU MIGHT HAVE A CURVE THAT LOOKS LIKE THIS. THERE'S F-OF X-. YOU MIGHT HAVE A CURVE THAT LOOKS LIKE THIS. G-OF X-. AND YOU MIGHT HAVE A CURVE THAT LOOKS LIKE THIS. I'LL CALL H-OF X-. YOU CAN IMAGINE THAT. AND WHEN I SAY THE AREA BOUNDED BY THE THREE CURVES. WELL, HERE IT IS. YOU MIGHT WANT THAT AREA. SO IT'S THE SAME IDEA THAT WE'VE DONE BEFORE. INTO THINGS WHERE WE CAN INTEGRATE. YOU HAVE TO BREAK IT UP SO HERE LET ME DRAW A

7 VERTICAL LINE THERE. LET ME CALL THAT A-AND CALL THAT B-AND CALL THAT C-. 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 7 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. IT'S SO THE TOP IS LITTLE F-. THE BOTTOM IS LITTLE G-. 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-. NOW THE TOP IS THIS CURVE. H-. AND THE BOTTOM IS STILL G-. 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. SO LET ME SAY WHAT I MEAN. SO LET ME DRAW A CURVE. E-TO THE MINUS X-. NICE SIMPLE CURVE. SUPPOSE I WANT THE AREA UNDER THIS CURVE. WHEN DO I STOP DRAWING RECTANGLES, IT GOES ON FOREVER. THE AREA UNDER THIS CURVE GOES ON AND ON

8 FOREVER. WELL WE DON'T KNOW HOW TO DO THAT YET. SO HELP STOP IT LITTLE B-. I WANT TO FIND THE AREA FROM ZERO, A-EQUALS ZERO OUT TO B-. 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 8 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. (ON BOARD). FROM ZERO OUT TO THIS NUMBER WHICHEVER ONE I PICK. NOW THE QUESTION IS WHAT HAPPENS AS I MOVE THIS OUT TO INFINITY. TAKE THE LIMIT. THAT'S THE QUESTION, WHAT IS THE AREA UNDER THIS ENTIRE CURVE ALL THE WAY OUT TO INFINITY. IT'S AN INFINITE REGION. 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-. HAPPENS TO E-TO THE MINUS B-AS B-GOES TO INFINITY. SO WHAT 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-. LET ME TAKE THE NOW WHAT I DRAW THAT WHAT'S IT LOOK LIKE? IT, IT HAS THE VERTICAL ASYMPTOTE AM GOES TO INFINITY. THAT'S WHAT THE CURVE LOOKS LIKE. AS X-GETS SMALL. SO I COULD ASK, FOR EXAMPLE, LET ME SAY FROM, I COULD ASK FOR THE AREA UNDER THE CURVE FROM A, SOME LITTLE POSITIVE

9 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. INTEGRAL FROM A-TO ONE, OF D-X-OVER THE SQUARE ROOT OF X-. SO THAT'S A NICE SIMPLE ONE. SO WHO CAN DO THE INTEGRAL OF X-TO THE MINUS 9 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-. SO WHAT'S THE LIMIT OF THAT? AS A-GOES TO ZERO? AS A-GOES TO ZERO THIS GOES TO ZERO SO IT'S JUST TWO. SO THE AREA UNDER THIS CURVE GOING ALL THE WAY OVER HERE IS JUST TWO. IT'S A PERFECTLY REASONABLE THING TO ASK. 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? SO THE LAST SECTION OF THE BOOK. SO WE'VE DONE A BUNCH OF APPLICATION OF

10 INTEGRATION THROUGHOUT THIS CHAPTER. SO NOW GOING TO DO SOME MORE. 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 10 LIKE A RIEMANN SUM. SO WHAT'S A RIEMANN SUM? IT'S A SUM WHERE THE SOME LITTLE GUY THERE, LITTLE DELTA X-. AND THESE X-OF I-S HAVE SOMEHOW SPACED OUT BETWEEN TWO NUMBERS A-AND B, EVENLY SPACED BETWEEN A-AND B-. AND YOU CAN RECOGNIZE WHAT WE WANTS LOOKS LIKE THIS. AND THEN, OF COURSE, THAT APPROACHES THE INTEGRAL. 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. SO LET'S JUST TRY AN EXAMPLE. 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

11 TO F-OF N-OVER F-WHICH IS ONE. TAKE N-DIFFERENT VALUES SPACED IN THERE. 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 11 REASONABLE THING. SO WHAT HAPPENS TO THIS AS N-GETS LARGER AND LARGER? THIS IS A AVERAGE FOR A PARTICULAR VALUE OF N-. DO YOU RECOGNIZE THAT AS A RIEMANN SUM. THAT'S EXACTLY WHAT THAT IS, IT'S A RIEMANN SUM. 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. IF I'M GOING FROM ZERO TO ONE. SO LET'S ASK WHAT ABOUT THE AVERAGE IF NOW I WANT TO TAKE,

12 I DON'T WANT TO GO FROM A-TO B-. I WANT TO FIND THE AVERAGE (INAUDIBLE). 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, 12 PLUS F-OF X-N-. SO THERE'S THE DEFINITION OF THE AVERAGE. SO THIS IS A RIEMANN SUM? ALMOST. 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-. IT'S THE INTEGRAL FROM A-TO B-DIVIDED INTO N-PIECES. 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. AND LET ME WRITE IT OVER HERE. SO WHAT I WANT IS THIS. (ON BOARD). I NEED TO WRITE THAT AS RIEMANN SUM. THERE IT IS. 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. THIS IS THE AVERAGE THAT I WANT. THERE'S A ONE OVER N-FACTOR. BUT TO

13 DO THE INTEGRAL, IT'S DELTA X-. AND TO GET ONE OVER N-I HAVE TO DIVIDE BY B-MINUS A-. DELTA X-OVER B-MINUS A- STUDENT: 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 13 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. IN 1998 WORLD POPULATION WAS 5.9 BILLION PEOPLE. OKAY. AND IT WAS GROWING AT THAT TIME AT 1.34 PERCENT PER YEAR. SO POPULATION IS GROWING. OKAY. SO THE QUESTION I'M GOING TO ASK IS WHAT IS THE AVERAGE WORLD POPULATION OVER 30 YEARS STARTING IN IN OTHER WORDS FROM 1998 TO I WANT TO KNOW THE AVERAGE POPULATION OVER THOSE 30 YEARS. 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 T EQUALS ZERO IS AND WHAT I WANT IS THE AVERAGE. SO THE AVERAGE IS GOING TO BE THE AVERAGE OVER 30 YEARS. SO I'M GOING TO GO FROM ZERO TO 30. TAKE THE POPULATION. AND INTEGRATE IT. BUT I HAVE TO PUT SOMETHING OUT FRONT. OVER HERE. WHAT AM I GOING TO DIVIDE BY?

14 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. MULTIPLIED BY SOMETHING. WHAT SHOULD I MULTIPLY IT BY? 5.9 BILLION. THE BEGINNING THE POPULATION IS 5.9. WHEN T EQUALS ZERO AT EVERY YEAR YOU ADD ONE TO T IT GROWS BY ABOUT 1.34 PERCENT. SO THE AVERAGE THEN IS, LET'S, I HAVE ONE 30TH AND I HAVE.. ZERO (ON BOARD). AND I NEED ONE MORE FACTOR TO MAKE THIS WORK. WHAT DO YOU NEED TO DO TO MAKE THIS WORK OUT RIGHT? ONE DIVIDED BY (ON BOARD). SO NOW YOU CAN PLUG IN 30. BE 7.26 BILLION. PLUG IN ZERO, SUBTRACT, AND IT COME OUT TO THAT'S THE NUMBER. STUDENT: ARE YOU MULTIPLYING BETWEEN THE 5.9 AND ONE -- PROFESSOR: 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). OKAY ANY QUESTIONS ABOUT AVERAGES? OKAY. SO THE LAST APPLICATION. SOLIDS OF REVOLUTION. SO I HAVE TO DEFINE WHAT I MEAN BY THAT. THE IDEA IS YOU TAKE A CURVE

15 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. THERE IS A WORD. SO HERE'S THE X-AXIS FROM A-TO B-. I'M GOING TO TAKE A REALLY REALLY SIMPLE CURVE, A STRAIGHT LINE. AND WHAT I'M GOING TO DO IS ROTATE THIS OUT OF PLANE. BREAK IT OUT LIKE THIS, AND THEN, AND MAKE A CIRCLE. 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. AND I GET A CYLINDER. SO LET ME JUST DO ANOTHER EXAMPLE. SUPPOSE I TAKE THIS CURVE, ALSO A STRAIGHT LINE BUT NOW IT GOES FROM ZERO UP TO SOMETHING. TO TAKE THIS THING AND ROTATE IT OUT OF BOARD. AND I'M GOING 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. THAT'S THE SHAPE I GET. AND ALL THE CROSS SECTIONS ARE CIRCLES. ALL THE THE CROSS SECTIONS. SO LET ME DO ANOTHER EXAMPLE JUST TO MAKE IT CLEAR. LET'S SUPPOSE I TAKE A SEMICIRCLE. NICE CURVE. AND ROTATE IT OUT OF THE PLANE. WHAT DO I GET P? A SPHERE. I GET A SPHERE. SO WRITE THIS DOWN. IF TAKE A SEMICIRCLE AND ROTATE IT YOU GET A SPHERE. IF YOU TAKE A HORIZONTAL LINE, YOU GET A CYLINDER. AND IF YOU TAKE A SLANTED LINE, YOU GET A CONE. AND LET ME JUST DO ONE MORE EXAMPLE TO MAKE IT CLEAR. SO THERE'S A-AND THERE'S B-. WHAT DO I GET? YOU GET A WINE GLASS. OKAY. SO YOU CAN GET ANY, SHAPE THAT YOU WANT. THAT'S THE GENERAL IDEA. SO WHY AM I TELLING YOU ALL THIS? THE QUESTION

16 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. HERE'S MY CURVE AND HERE'S ANY SOLID REVOLUTION. SO LET ME REMIND OURSELF THE STEPS TO GET THE 16 AREA UNDER OF CURVE. TO GET MY RIEMANN SUM. I IMAGINE ALL OF THESE LITTLE RECTANGLES 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-. THE WHOLE AREA, SO THAT'S ONE AREA. AND THE WHOLE AREA, OF COURSE, IS JUST ADD THEM ALL UP. THIS IS JUST REVIEW. AND LOW AND BEHOLD I GET THIS INTEGRAL. THAT'S WHAT WE DID FOR AREAS. I WANT GOING TO DO THE SAME THING FOR VOLUMES. 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. BREAK IT UP INTO LOTS OF LITTLE SLICES THAT LOOK LIKE THIS. (ON BOARD). LITTLE THIN CYLINDRICAL SLICES. WHAT I'M GOING TO DO IS TAKE ANY ONE OF THESE, IT'S GOING TO LOOK LIKE A LITTLE FLAT DISK. WHAT I WANT TO DO NOW, AND IT HAS A RADIUS. WHAT'S THE RADIUS? THE RADIUS IS F-OF X-OF I-. THAT POINTS RIGHT THERE IS X-OF I-.

17 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. SO IT'S GOING TO BE DELTA X-TIMES WHAT? WHAT'S THE AREA OF THE CIRCLE AND THE BASE? IT'S PI TIMES F-OF X-OF I-SQUARED. THAT'S JUST THE AREA OF THIS CIRCLE, THE BASE. 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. AND SO NOW WHAT HAPPENS IS I LET N-GOES TO INFINITY? THIS LOOKS LIKE A RIEMANN SUM, RIGHT? IT'S DELTA X, THERE'S A PI THAT FACTOR OUT, DELTA X-TIMES SUM OF SOME FUNCTION EVALUATIONS. SO LET ME PULL OUT THE PI. IT'S GOING TO BE A INTEGRAL FROM A-TO B-. I SHOULD HAVE SAID FROM A-TO B-. FROM A-TO B-. BUT WHAT'S THE FUNCTION WHOSE VALUES I'M ADDING UP? F-OF X-SQUARED. THERE IT IS. THERE'S THE ANSWER FOR THE VOLUME. A NICE SIMPLE FORMULA. JUST INTEGRATE F-SQUARED INSTEAD OF F-AND MULTIPLY BY PI. STUDENT: WHAT ABOUT DELTA X- PROFESSOR: THAT HAS PART THE RIEMANN SUM. REMEMBER WE LET N-GO TO INFINITY. DELTA GETS SMALLER AND SMALLER AND THE WHOLE THING CONVERGES TO THE INTEGRAL. THE SAME THING HAPPEN OVER HERE. NOW VOLUMES ARE EASY TOO. LET'S DO SOME. SO I'M GOING TO USE

18 THESE EXAMPLES. SO LET ME MAKE THIS GO FROM ZERO TO H-. BE THE LENGTH, THE HEIGHT OF THE CYLINDERS. SO H-IS GOING TO AND LET ME LET THIS DISTANCE BE R, THE RADIUS. 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-. AND WHAT'S THE INTEGRAL OF ONE? (ON BOARD) IT'S JUST X-. AND THAT'S PIE R-SQUARED H-. DOES THAT LOOK FAMILIAR I HOPE? 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. LET ME DO A CONE. 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-. AND LET'S SUPPOSE THE RADIUS OF BASE OF CONE IS R-. SO IT'S GOING TO BE A CONE OF HEIGHT H-AND RADIUS R-AT THE BASE. I WANT TO FIND ITS VOLUME. TO DO THAT I NEED, THIS DISTANCE HERE IS GOING TO BE R-. FUNCTION THAT GIVES ME THIS STRAIGHT LINE. WHAT IS THE WHO CAN TELL ME WHAT STRAIGHT LINE GOES THROUGH THE ORIGIN AND THROUGH POINT R-. SO THIS POINT HERE IS R-COMMA H-. H-COMMA R, SORRY. GOES OUT

19 H-UP R-. WHAT, SO WHAT'S THE FUNCTION GIVES ME THAT STRAIGHT LINE? EQUATION OF A STRAIGHT LINE THERE? IT'S X-TIMES THE SLOPE, SO WHAT'S THE SLOPE OF THIS LINE? 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). SEE WHAT WE GET. SO THESE R-AND H-S ARE CONSTANT. FACTOR THOSE OUT FRONTS. R-SQUARED OVER 19 H-SQUARED. I'M LEFT WITH X-SQUARED. SO WHAT'S THE INTEGRAL OF X-SQUARED? IT'S X-CUBED OVER THREE, EVALUATED AT H-AND ZERO. SO I PLUG IN H-. I GET H-CUBED OVER THREE, PLUG IN ZERO I GET ZERO. SO I GET PI R-SQUARED H-SQUARED H-CUBED OVER THREE. SOMETHING CANCEL HERE, RIGHT? SO I GET PI OVER THREE R-SQUARED H-. THERE IT IS. STUDENT: WHERE DO WE GET R-OVER H-X- PROFESSOR: SO R-OVER H-IS THE EQUATION OF THIS LINE. SLOPE OF THE LINE WHICH IS THE BOUNDARY OF THE CONE. IT IS THE BECAUSE THE CONE GOES OUT TO H-. AND IT GOES UP TO R-. SO THAT'S WHERE IT COMES FROM. SO DID I GET AT THAT RIGHT? PI OVER THREE R-SQUARED H-. 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. SO WRITE DOWN X-SQUARED PLUS Y-SQUARED EQUALS R-SQUARED. I KNOW

20 THAT'S HOW YOU DEFINING A SEMICIRCLE AM IF I SOLVE FOR Y-I GET Y-IS THE SQUARE ROOT (ON BOARD). SO THIS IS F-OF X-. (ON BOARD). SO THAT TELLS ME I SHOULD COME HERE AND DO THE INTEGRAL OF PI, SO WHAT ARE THE LIMIT OF INTEGRATION? FOR THE SPHERE WHAT DO I INTEGRATE FROM? I WANT TO INTEGRATE FROM ALL THE WAY OVER HERE TO ALL THE WAY OVER THERE WHICH IS FROM WHERE TO WHERE? NOT FROM ZERO. FROM MINUS R-TO R-. SO INTEGRATE FROM MINUS R-TO R-. 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-. LET'S JUST DO IT. THE INTEGRAL OF R-SQUARED IS R-SQUARED X-. AND INTEGRAL OF MINUS X-SQUARED IS MINUS X-CUBED OVER THREE. I EVALUATED AT M, EVALUATED AT MINUS R-AND SUBTRACT. WHEN ALL IS SAID AND DONE WHAT YOU THINK THE ANSWER IS? ANYBODY? WHAT'S THE VOLUME OF SPHERE? FOUR THIRDS PI R-CUBED. THAT'S WHERE IT COMES FROM. 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. AND SO IT'S GOING TO BE A SQUASHED CIRCLE. SO IT'S GOING TO GO OUT TO A-IN THAT C AND B-IN THAT DIRECTION. CALL AN ELLIPSOID. 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

21 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. I GET THAT Y-IS THE SQUARE ROOT OF B-SQUARED (ON BOARD). 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. AND WHAT ARE THE LIMIT OF INTEGRATION FROM WHERE TO WHERE? NEGATIVE A-TO A-OF F-OF X SQUARED. 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. SO THAT CANCELS. (ON BOARD). NOW I'LL INTEGRATE TERM BY TERM, WHAT IS 21 THE INTEGRAL OF B-SQUARED, B-SQUARED TIMES X-. INTEGRAL OF X-SQUARED? IT'S X-CUBED OVER THREE. AND WHAT IS THE 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-. SO A NICE SIMPLE FORMULA. SO IF I COME BACK HERE, LET'S DO A SANITY CHECK. 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-. IF B-EQUALS A-I GET FOUR THIRDS PI TIMES RADIUS CUBED. I GET THE SAME THING I GOT OVER THERE. YOU SHOULD, THAT GIVES YOU A NICE WARM FEELING YOU DID THE MATH RIGHT. SO IT DOESN'T CHANGE VERY MUCH TO GET THIS VOLUME. OKAY. THAT'S THE END OF THE COURSE EXCEPT FOR REVIEW. ANY QUESTIONS? STUDENT: DO YOU HAVE OFFICE HOURS TOMORROW. PROFESSOR: YES. I APOLOGIZE I HAVE A SICK CHILD AT HOME ON

22 MONDAY. I'LL BE BACK. SO WE HAVE A CHOICE NOW. A THREE WAY CHOICE. I CAN DO SAMPLE PROBLEMS FROM THE FINAL. 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: I'M DOING THESE AS A EXAMPLES SO YOU KNOW HOW TO DO THEM. 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. YOU HAVE TO KNOW ABOUT NATURAL LOGARITHMS. 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. YOU BETTER KNOW HOW TO DIFFERENTIATE IT TOO. 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

23 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: DO WE ONLY HAVE TO KNOW. PROFESSOR: 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 IT'S COMPOUNDED CONTINUALLY IT'S AN EXPONENTIAL FUNCTION. 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. AND THEN YOU CAN WRITE DOWN THE ANSWER. 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

24 HALF OF WHAT YOU STARTED WITH. THE HALF LIFE IS WHEN HALF OF IT DISAPPEARS. 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-. (ON BOARD). DECAY. 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. WITH DIFFERENT MODEL FOR HOW DO YOU APPROACH LIMITS. EXPONENTIALLY IS CERTAINLY ONE WAY. AND WE ALSO LOOKED AT A VARIATION ON IT CALLED THE LOGISTIC CURVE. SO THESE WERE MODEL FOR DIFFERENT THING. 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. STARTING AT ZERO. 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. WHEN Y-GETS CLOSE TO -- IT GETS FLATTER AND FLATTER. THE DERIVATIVE, IF YOU CAN FIGURE OUT FROM

25 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. THE OTHER ONE THAT WE USED ALSO HAD A FINAL LIMIT VALUE M. BUT IT COULD START ANYWHERE. 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. SO HERE, HERE IT IS. THERE IS THE RATE. 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. AND YOU CAN, THAT ALSO HAS TO BE GIVEN. 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

26 ON M. AND THERE'S SOME ALGEBRA FOR FIGURING IT OUT BUT I WON'T REPEAT IT. STUDENT: IS IT ONE PLUS B-OR ONE MINUS B- PROFESSOR: SORRY, 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. THEN WE TALKED ABOUT ANTI DIFFERENTIATION. FOR EVERY RULE OF DIFFERENTIATION THERE WAS ONE THAT WORKED BACKWARDS. YOU SHOULD KNOW ALL OF THEM, THESE, BY HEART. SO THIS IS, DIFFERENTIATE THIS, THE CONSTANT GOES AWAY, YOU GET BACK TO THAT. THIS DOES NOT WORK WHEN ARTICLE IS MINUS ONE. 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. SO INTEGRAL, F-OF X-PLUS OR MINUS G-OF X-. 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

27 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. AND THAT GOES FROM THE INTEGRAL FROM A-TO B-OF F-OF X-D-X-. THAT'S WHAT THE DEFINITE INTEGRAL MEANS. 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: IF THE M FUNCTION WHAT'S UNDERNEATH THE SUM. PROFESSOR: 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-. IT'S JUST SHORTHAND NOTATION FOR THAT THING THERE. 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. SO THESE TWO OPERATIONS CANCEL ONE ANOTHER OUT. SO THAT'S ANOTHER WAY TO SAY WHAT THE FUNDAMENTAL THEOREM OF CALCULUS IS. ALMOST DONE HERE. YOU SHOULD BE ABLE TO DO AREA PROBLEMS. WE'VE DONE A BUNCH OF THOSE ALREADY. THIS FUNCTION IS G OF X-. SO IF THIS FUNCTION IS F-OF X, AND AND YOU WANT TO FIND THE AREA IN BETWEEN A-AND B, THEN (ON BOARD), OF COURSE, THERE ARE

28 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. SO YOU KEEP INTEGRATING. AND THE AVERAGE VALUE OF THE FUNCTION. AND WHAT WE JUST DID EARLIER, VOLUMES OF SOLIDS. OF REVOLUTION. OKAY. LOTS OF DIFFERENT EXAMPLES OF THAT. AND I THINK THAT'S A GOOD PLACE TO STOP. 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).

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