1. Lesson 5-4b
Net Change Theorem

2. Icebreaker
Evaluate and explain the meaning of: (3 - x) dx sin (x) dx 6
= 3x - ½x² + c | 6 ∫
= [3(6) - ½6² + c] – [3(0) - ½0² + c]
= [18 – 18 + c] – [ c] = 0 2π
= cos(x) + c | 2π ∫
= [cos(2π) + c] – [cos(0) + c]
= [1 + c] – [1 + c] = 0

3. Objectives
Solve indefinite integrals of algebraic, exponential, logarithmic, and trigonometric functions
Understand the Net Change Theorem
Use integrals to solve distance problems to find the displacement or total distance traveled

4. Vocabulary
Indefinite Integral – is a function or a family of functions
Distance – the total distance traveled by an object between two points in time
Displacement – the net change in position between two points in time

5. Example Problems
A deer population is increasing at a rate of dp/dt = t per year (where t is measured in years). By how much does the deer population increase between the 4th and 10th years? Show integral set up and answer.
P = ( t) dt ∫ 4 10
P = 20t + ½ (35)t² + C | 10 4
[20(10) + ½ (35)(10)² + C] - [20(4) + ½ (35)(4)² + C]
[ C] - [ C] = increase in deer

6. Displacement vs Distance Traveled∫ t2 t1
displacement = v(t) dt = A1 – A2 + A3
distance = |v(t)| dt = A1 + A2 + A3 ∫ t2 t1 v A3 A1 t t1 t2 A2
displacement is net change in position
distance is the total distance traveled

7. ∫ Example Problems cont Evaluate:
2) (1 – x²) dx ∫ -1 2
= (x – ⅓x³) | = [2 - 8/3] - [1 – 1/3] -1 2
= [-2/3] - [2/3] = -4/3

8. (x – ⅓x³) | - (x – ⅓x³) | = [2/3 - -4/3] - [-2/3 – 2/3]
Example Problems cont
Remember definition of |x| !
x if x > 0
F(x) = |x| =
-x if x < 0
Evaluate:
3) |(1 – x²)| dx ∫ -1 2
(1 – x²) < 0 if x > 1 SO
(1 – x²) dx (1 – x²) dx ∫ -1 1 2
(x – ⅓x³) | (x – ⅓x³) | = [2/3 - -4/3] - [-2/3 – 2/3] -1 1 2
[6/3] - [-4/3] = 10/3

9. ∫ ∫ Example Problems cont
Problem 55 from the book: Given the following: A(t) = t + 4, v(0) = 5, 0 ≤ t ≤ 10 Find the velocity at time t Find the distance traveled during the given interval
V(t) = (t + 4) dt = ½t² + C V(0) = 5 = C ∫
S(t) = (½t² + 5) dt = (1/6)t³ + 5t + C |
= (1000/ C) – (C) = ∫ 10

10. Summary & Homework Summary: Homework: Definite Integrals are a number
Evaluated at endpoints of integration
Indefinite Integrals are antiderivatives
Homework:
Day One: pg : 19, 22, 27, 28,
Day Two: pg : 3, 7, 9, 61 (see appendix E)