Calculus AB, Assignment Six

We have now studied the definite integral. What we haven't done is seen how it is useful to us in any applied setting. That will be remedied here because we will apply it in several ways, many of which are downright fun (volumes....). A basic principle will be determined that tells us that the definite integral of a rate of change of a quantity will equal the net change in the quantity over that interval. We will also marry the concepts of taking derivatives (and antiderivatives) to the concept of definite integral. The FUNDAMENTAL THEOREM OF CALCULUS, one of the most powerful discoveries in Calculus ties together these two seemingly independent ideas. By using the FTC 1, we discover we no longer need to take limits to infinity of Riemann Sums. We will investigate fully how the integral of a function from a fixed point to a variable endpoint (known as an accumulation function) and the derivative of that accumulation function are related to the integrand function within(FTC 2).

Day Number

Class Topic 

Homework
Reading Class notes and Examples is part of EVERY NIGHT'S homework. 

One

The Indefinite 
Integral: 
Introduction to antidifferentiation and integral curves. [fnint(f(x),x,x)] 

Pg 389# 1,3,7-19,odd,43 
Pg 208 (not a mistake) #1,5,11,15

Two

Integration using the method of U-substitution

Pg 390 # 25-29odd, 35 Pg 395 # 1,9,13,15

Three

More integrating

Pg 395#17-27 odd,49

Four

Fundamental Theorem of Calculus: (FTC Part I): Statement/Proof and applications

Pg 395 # 31,33,55 Pg 425 # 1-7, 11-19 odd

Pg444#1,3,13,15,19

QUIZ In 2 classes.

Five

Mean (average) value theorem for integrals and upper and lower bounds for definite integrals

Pg 425 # 1,9,21,23,25,29,41,55,57,59.

Six

Rectilinear motion revisited-intro

 Pg 437-440 #7,9, 49, 51 ,55. 57 

Seven

Applications (including rectilinear motion). Interpreting area under the curve of a derivative function as amount of change in a quantity.

 Pg 437-440 #1,3 (answers in book are in error, see below), 5,15,17,19

Answers: 3a) disp=-.5, dist=1.5; b) disp=1.5, dist=2
 

Eight

Accumulation functions

 Pg 437-440 #11, 53, 61 and Handouts (details in class). Read 446-447: Ln and integrals then do Pg 451 #1. 

Nine

FTC Part 2: The inverse relationship between differentiation and integration.

 Pg 426-427 #43-49 odd and Handouts (details in class). Pg 452 #23,27.

Ten

 Intro to Area between curves in X and Y

 Pg 467 #1, 7, 9, 19 and Handouts. Pg 452#25 (see #23).

Eleven

Volumes with similar cross sections

 Pg 467 # 3, 5, 11, 13 , 29 and Handouts. Pg 474 #33, 35 a) b), Handouts

Twelve

Intro to Volumes of revolution: Disks/Washers

 Pg 467 # 17, 21. Pg 474 #5,7,9, 15,17, 35c)

Thirteen

More Volumes of Revolution using Disks/Washers

 Pg 474 #11,13, 21, 25, 27, 29.

Fourteen

 AP Exam Problems. 
 
 

 

 Pg 454-455 #2, 4, 7,9,11,19,27,29. 31 

Fifteen

Volumes using Shells

 Pg. 1-7 odd,13, 17,21

Sixteen

More?

Pg. 9, 11, 15, 19

Fifteen

EXAM