1 MATH 16 A-LECTURE. DECEMBER 4, 2008. PROFESSOR: WELCOME BACK. TWO MORE LECTURES. BEFORE THE HOLIDAYS SO WE'RE ALMOST DONE WITH THE BOOK. I APOLOGIZE FOR NOT POSTING THE HOMEWORK YET. I'VE BEEN STRUGGLING TO CATCH UP EVERY SINCE SO TONIGHT I WILL POST THE HOMEWORK. IT WAS A FUN MEETING BUT I WASN'T PLANNING ON STAYING. SO LET'S GO ON WITH INTEGRATION. LET ME JUST REVIEW THE MOST IMPORTANT FACTS FROM SECTION 6.1 AND SECTION 6.2 THAT DANIELLE DID TWO DAYS AGO. AND HERE IT IS. SO IF F-OF X-IS GIVEN AND CAPITAL F-SATISFIES THE PROPERTY THAT ITS DERIVATIVE OF BIG F-GIVES YOU LITTLE F-THEN BIG F-IS CALLED THE ANTI DERIVATIVE. ANTI BECAUSE IT'S GOING BACKWARDS. AND WE HAD A NOTATION FOR THAT. AND IT WAS THIS SQUIGGLY S-. WHICH IS THE INTEGRAL SIGN. WE WROTE IT THIS WAY. C-IS ANY CONSTANT. THIS SAYS THAT THE ANTI DERIVATIVE OF LITTLE F-IS BIG F-PLUS ANY CONSTANTS IF WE DIFFERENTIATED SIDES THE CONSTANTS GOES AWAY AND I GET F-OF THAT, THE THING TO NOTE HERE IS IF YOU TAKE BOTH SIDES OF THIS AND DIFFERENTIATE THEM, JUST GET D-D-X-OF F-OF X-AND WE SAID THAT WAS F-OF X-. SO IF YOU LIKE THESE TWO SYMBOLS CANCEL ONE ANOTHER OUT. THEY'RE OPPOSITES OF ONE ANOTHER. SO JUST TO REVIEW. AND SO AS A RESULT OF THAT, FOR EACH DIFFERENTIATION RULE WE EVER LEARNED, AND THERE WERE A FEW OF THEM, YOU GET INTEGRATION RULE BY USING IT BACKWARDS. AND THERE ARE A LOT OFTHEM SO I DON'T WROTE THEM ALL DOWN AGAIN. BUT ONE EXAMPLE, X-TO THE R, THAT GIVES US THIS BECAUSE WHEN YOU DIFFERENTIATE 2 THIS THE R-PLUS ONE COMES DOWN AND CANCELS AND YOU GET R-IN THE DENOMINATOR. BUT THERE IS ONE ACTUAL RESTRICTIONS ON THIS RULE. WHAT CAN R-NOT EQUAL? R-DOES NOT EQUAL NEGATIVE ONE, BECAUSE THEN THIS WON'T MAKE SENSE. BUT THAT IS ONE OF THE SEVERAL RULES, GOOD TO KNOW. SO THAT'S 6.1. JUST TO REVIEW. SO THEN 6.2, WAS ABOUT AREAS. UNDER CURVES. SO THE IDEA WAS WE'RE NOW GOING TO LEARN A WAY TO FIND OUT IF SOMEBODY GIVES YOU A NICE CURVE THERE, AND LET'S KEEP THE PICTURE EASY AND ASSUME F-OF X-IS POSITIVE, YOU WANT TO FIND THE AREA UNDER OF CURVE. SO THIS WHOLE AREA, UNDERNEATH F-ABOVE THE AXIS AND BETWEEN A-AND B, EASY WAY TO DO IT, TO DEFINE WHAT IT MEANS TO HAVE AN AREA HERE WHERE THE TOP CAN BECOME SOME SQUIGGLY THING, IS IS YOU BREAK UP THE AREA INTO LOTS OF TALL SKINNY REGIONS. SO THE SUM OF THE AREA OF THE REGIONS IS THE AREA OF THE WHOLE THING MUCH WE'RE GOING TO APPROXIMATE EVERY ONE OF THESE LITTLE REGIONS WITH A RECTANGLE BECAUSE WE KNOW HOW TO FIND AREAS OF RECTANGLES. AND SO HERE'S ONE EXAMPLE OF HOW YOU WOULD APPROXIMATE THAT BY A BUNCH OF AREAS OF RECTANGLES. SO WHAT IS THE AREA OF ONE OF THESE RECTANGLES IF, LET ME CALL THIS POINT HERE, SO LET ME DIVIDE UP ALL THESE RECTANGLES, ARE GOING TO HAVE THE SAME WIDTH. THE WIDTH IS I'M GOING TO CALL DELTA X-. AND I'M GOING TO HAVE N-OF THEM, SO N-RECTANGLES. ALL OF THE SAME WIDTH. EACH ONE I'M GOING TOPICK A POINT. SO, FOR EXAMPLE, THIS POINT RIGHT THERE I'M CALL X-ONE? THAT POINT THERE I'LL CALL X-TWO. WHAT IS THE HEIGHT OF THE RECTANGLE? THAT HEIGHT IS F-OF X-ONE. THAT HEIGHT RIGHT 3 THERE IS F-OF X-TWO AND SO FORTH. SO THE AREA OF THE ITH RECTANGLE IS GOING TO BE THE BASE TIMES HEIGHT. BASE IS DELTA X, THE HEIGHT IS THE HEIGHT OF THE RECTANGLE, F-OF X-OF I-. AND THE AREA OF ALL THE RECTANGLES IS JUST SUM ALL THAT UP. SO IT'S THE SUM FROM I-EQUAL ONE, OF (ON BOARD). AND NICE SIMPLE SUM. AND THERE'S A NAME FOR THAT. NAMED AFTER THE PERSON WHO INVENTED THIS IDEA, RIEMANN SUM. I WANT TO FIND THE AREA OF THIS CURVE, APPROXIMATE IT BY BREAKING UP INTO LOT OF RECTANGLES BECAUSE I KNOW HOW TO FIND AREAS OF RECTANGLES, AND THE EXACT AREA IS GOTTEN BY TAKING, MAKING THESE RECTANGLES SMALLER AND SMALLER AND TAKING THE LIMIT. SO IT'S GOING TO BE THE RIEMANN SUM, TAKE DELTA X-OUT FRONT. FACTOR OUT. SO BREAK IT UP RECTANGLES INTO DELTA X-. GOING TO BE N-OF THEM AND LET THE NUMBER OF RECTANGLES GET BIGGER AND BIGGER AND THE SUM OF ALL THE RECTANGLE AREA IS GOING TO CONVERGE TO THE AREA UNDER OF CURVE. SO THAT'S THE POINTS OF 6.2. HAVEN'T TOLD YOU HOW TO COMPUTE IT YET BUT THAT'S HOW YOU FIND THE AREA UNDER THE CURVE. BREAK UP INTO LITTLE RECTANGLES MORE AND MORE. SO THAT WAS THE IDEA. NOW THAT PICTURE MAKES SENSE I HOPE. WHEN F-OF X-WAS POSITIVE BUT THE QUESTION IS WHAT HAPPENS WHEN F-OF X-IS NEGATIVE. MAKESURE WE UNDERSTAND THE DEFINITION. THIS IS OKAY, ALL MAKES SENSE IF F-OF X-IS POSITIVE. WHAT IF F-OF X-IS NEGATIVE? SO THE MAIN IDEA IS WE'RE STILL GOING TO USE THE RIEMANN SUM. THIS IS THE THING WE'RE GOING TO KNOW HOW TO COMPUTE. SILL GOING TO USE THIS. BREAK UP INTO LOTS OF RECTANGLES. AND SUM UP ALL THE 4 AREAS OF RECTANGLES. AND TAKE THAT LIMIT. BUT WHAT IF F-OF X-IS NEGATIVE? THEN MY SUM CAN BE NEGATIVE, SO THE ANSWER IS GOING TO BE, WHEN YOU ASKED THE QUESTION WHAT'S THE AREA, IT'S GOING TO BE NEGATIVE, WHAT IS THAT? LET ME DRAW SOME PICTURES. SO LET ME TAKE NICE SIMPLE CURVE Y-EQUALS MINUS X-. BETWEEN ZERO AND ONE. SO WHAT IS THE, IF YOU DO THE RIEMANN SUMS WHAT DO YOU GET, YOU GET SOMETHING THAT LOOKS LIKE THIS. YOU KNOW