CALCULUS AP BC Q301CH5A: (Lesson 1-A) AREA and INTEGRAL Area Integral Connection and Riemann Sums

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1 CALCULUS AP BC Q301CH5A: (Lesson 1-A) AREA and INTEGRAL Area Integral Connection and Riemann Sums

2 INTEGRAL AND AREA BY HAND (APPEAL TO GEOMETRY) I. Below are graphs that each represent a different f() from = -3 to = 4. A] Find the Area bounded by the graph of f and the -ais. B] Evaluate the integral 4 3 f ( d ). Epress the integral as it relates to a collection of areas. C] Epress the Area as it relates to an (or set of) integral(s).

3 II. Evaluate each integral by hand (appealing to geometry). 3 A] 1 1 d B] 4 d 3 C] d 6 9 D] 3 5 d

4 CH5 (INTEGRAL PROPERTIES): Suppose that f and h are continuous functions and that 1 ) ( 9 1 d f, 5 ) ( 9 7 d f, and 4 ) ( 9 7 d h. Find each integral below: 9 1 ) ( d f 9 7 ) ( ) ( d h f 9 7 ) ( 3 ) ( d h f 1 9 ) ( d f 7 1 ) ( d f 7 9 ) ( ) ( d f h

5 HW: INTEGRAL AND AREA BY HAND (APPEAL TO GEOMETRY) Section 5.: 7, 13, 15, 17, 7, 39 INTEGRAL PROPERTIES: Section 5.3: 1, 5, 47 RIEMANN SUM and PROPERTIES: Section 5.: 1, 5, 47, 48, 55 CH5A Lesson 1-A Review (attached):

6 AP CALCULUS: Q301 CH5A Lesson 1-A Review A1 A3 A In the diagram above, the values of the areas A1, A, and A3 bounded by the graph of f () and the -ais, are 7, 5, and 8 square units respectively. f () has zeros at -4, -0.6, 3, and 6.5. Calculate the following definite integrals: f ( ) d 6.5. f ( ) d f ( ) d f ( ) d f ( ) d f ( ) d

7 CALCULUS AP BC Q303CH5A: (Lesson 1-B) AREA and INTEGRAL FTC: Fundamental Theorem of Calculus (Part II) Evaluation Method IA. [FTC] Evaluate the following integrals (a) BY HAND and (b) BY TI89

8 IB. [FTC-UB]: (No Calculator) Rewrite and evaluate each integral using the appropriate u-substitution and u-bounds d. sin( 5 ) d 0

9 II. [FTC-AA] Find the area bounded by y 3 3 and the -ais on the interval [0, ] (a) BY HAND (b) BY TI89

10 III. [FTC ID]: (Technology Required) Fundamental Theorem of Calculus in disguise 1. Suppose f / ( ) e and f ( 1) 3. Find f ().. Suppose / cos( ) f ( ) ( e ) and ( ) 1 f. Find f (0).

11 IV. [FTC-AV]: AVERAGE VALUE: 1. (No Calculator) Find the average value of f ( ) 3 3 on [0,]. (No Calculator) Find the average value of f ( ) sin on, 6

12 HW: INTEGRAL FTC BY HAND: Section 5.3: 1, 3, 5, 7 Section 5.4: 7, 9, 31, 33, 35, 37 INTEGRAL FTC BY TI89: Section 5.4: 49, 50 AREA FTC BY TI89: Section 5.4: 41, 43, 45, 47 AVERAGE VALUE BY HAND (APPEAL TO GEOMETRY): Section 5.3: 15, 16 AVERAGE VALUE BY HAND (FTC): Section 5.3: 3, 34, 35 AVERAGE VALUE BY TI89: Section 5.3: 11, 13 FTC-UB 1. Rewrite and evaluate cos( ) d using an appropriate u-substitution and u-bounds Rewrite and evaluate d using an appropriate u-substitution and u-bounds. FTC-ID 1. Suppose. Suppose ( sin and ( 1) 5 / f ( ) ) ( ln and ( ) 5 / f ( ) ) f. Find f (4) f. Find f (0.5)

13 Q301.CALCULUS AP BC Chapter 5 Lesson : FTC Part Applications FTC: Fundamental Theorem of Calculus (Part II) Lesson : APPLICATIONS A. Area Connection (Lesson 1) B. Evaluation Method [FTC] (Lesson 1) C. FTC APPLICATIONS C1: Displacement/Position/Total Distance C: Accumulating Quantity/(Rate In Rate Out)

14 C1: Displacement/Position/Total Distance A particle moves along the -ais such that its position at time t is given as (t). / The velocity of the particle at time t is given as ( t) v( t). Review: The particle is moving to the right when v ( t) 0 / Acceleration is positive when v ( t) 0 The particle is getting faster (speed increasing) when v (t) and a(t) share the same sign. New: C: Accumulating Quantity/(Rate In Rate Out)

15 NOTES 1. An object moves along the -ais with initial position ( 0). The velocity of the object at time t 0 is given by v( t) sin t. 3 A. What is the total distance traveled by the object over the time period 0 t 4? B. What is the position of the object at time t = 4? C. What is the average velocity over the time period 0 t 4? D. What is the average acceleration over the time period 0 t 4?. A particle moves along the -ais so that its velocity at time t is given as t v ( t) t 1 sin. At time t = 0, the particle is at position = 1. A. What is the total distance traveled by the particle from time t = 0 until time t = 3? B. What is the position of the particle at time t = 3? C. What is the average velocity from time t = 0 until time t = 3? D. What is the average acceleration from time t = 0 until time t = 3? 3. AP:000#

16 4. The rate at which people enter an amusement park on a given day is modeled by the function E defined by E ( t). t 4t 160 The rate at which people leave the same amusement park on the same day is modeled by the 9890 function L defined by L ( t). t 38t 370 Both E(t) and L(t) are measured in people per hour and time t is measured in hours after midnight. These functions are valid for 9 t 3, the hours in which the park is open. At time t = 9, there are no people in the park. A. How many people have entered the park by 5:00 pm (t = 17)? Round to the nearest whole number. B. The price of admissions to the park is $15 until 5:00 pm. After 5:00 pm the price of admissions to the park is $11. How many dollars are collected in admissions to the park on the given day? Round to the nearest whole number. C. Find H(17). D. Write a function H(t), the number of people in the park at time t. 5. Traffic flow is defined as the rate at which cars pass through an intersection, measured in cars per minute. The traffic flow at a particular intersection is modeled by the function F defined by t F ( t) 8 4sin for 0 t 30, where F(t) is measured in cars per minute and t is measured in minutes. A. To the nearest whole number, how many cars pass through the intersection over the 30- minutes period? B. What is the average traffic flow over the time interval 10 t 15? Indicate units of measure. C. What is the average rate of change of the traffic flow over the time interval 10 t 15? Indicate units of measure.

17 HW 1. A particle moves along the y-ais so that its velocity v at time t 0 is given by 1 t v( t) 1 tan e. At time t = 0, the particle is at y 1. A. Find the total distance traveled by the particle between time t = 0 and time t =. B. What is the position of the particle at time t =? C. What is the average velocity over the time period 0 t? D. What is the average acceleration over the time period 0 t?. AP:011#1 3. AP:009#1 4. The tide removes sand from Sandy Point Beach at a rate modeled by the function R, given by 4 t R( t) 5sin. 5 A pumping station adds sand to the beach at a rate modeled by the function S, given by 15t S( t). 1 3t Both R(t) and S(t) have units of cubic yards per hour and t is measures in hours for 0 t 6. At time t = 0, the beach contains 500 cubic yards of sand. A. How much of the sand will the tide remove from the beach during this 6-hour period? Indicate units of measure. B. Write an epression for Y(t), the total number of cubic yards of sand on the beach at time t. C. Find the rate at which the total amount of sand on the beach is changing at time t = 4. D. Find the total number of cubic yards of sand on the bank at time t = AP:010#1 6. AP:010#3

18 Q301.CALCULUS AP BC Chapter 5 Lesson 3: Fundamental Theorem (Part 1) Fundamental Theorem of Calculus (Part I)

19 APPLICATION (1, 4) (, 1) - (4, -1) - The graph of the function f, consisting of three line segments, is given above. Let g ( ) f ( t) dt. A. Compute g(4) and g(-) B. Find the instantaneous rate of change of g, with respect to, at = 1. C. Find the absolute minimum value of g on the closed interval [-, 4]. Justify your answer. D. The second derivative of g is not defined at = 1 and =. How many of these values are - coordinates of points of inflection of the graph of g? Justify your answer. 1

20 Fundamental Theorem of Calculus (Part I) PROOF

21 Fundamental Theorem of Calculus (Part II) PROOF

22 Lesson 3 HW: AP 004 #5 AP 005 FB #4 AP 007 FB #4 AP 008 FB #5 Section 5.4 #7 19odd, 61

23 Q301.CALCULUS AP BC Chapter 5 Lesson 4: Approimations of the Definite Integral Approimating a Definite Integral with a Riemann Sum RECTANGLE APPROXIMATION b n k 1 f ( d ) f ck k on ab, where f ( c k ) is the value of f at = c on the kth interval. a TRAPEZOIDAL APPROXIMATION b f ( d ) y0 y1 y yn 1 yn where y f( ) and is constant. a

24 1. Consider the area under the curve (bounded by the -ais) of f ( ) from 1 to 5. Use 4 equal rectangles whose heights are the left endpoint of each rectangle to approimate the area. (LRAM) Use 4 equal rectangles whose heights are the right endpoint of each rectangle to approimate the area. (RRAM) Use 4 equal rectangles whose heights are the midpoint of each rectangle to approimate the area. (MRAM) Use 4 trapezoids of equal width to approimate the area. (TRAM)

25 3.5. Use LRAM, MRAM, and TRAM with n = 3 equal rectangles to estimate the f ( ) d where values of the function y f () are as given in the table below y Use a Trapezoidal approimation with n = 3 trapezoids to estimate the f ( ) d where values of the function y f () are as given in the table below y

26 4. t (hours) R(t) (gallons per hours) The rate at which water flows out of a pipe, in gallons per hour, is given by a differentiable function R of time t. The table above measured every 3 hours for a 4-hour period. 4 A. Using correct units, eplain the meaning of the integral R( t) dt in terms of water flow. 0 4 B. Use a trapezoidal approimation with 4 subdivisions of equal length to approimate R ( t) dt. 0 C. Use a midpoint Riemann sum with 4 subdivisions of equal length to approimate the average of R(t). 1 D. The rate of water flow R(t) can be approimated by Q( t) 768 3t t. Use Q(t) to 79 approimate the average rate of water flow during the first 4-hour time period. Indicate the units of measure. HOMEWORK: # FormB # #