1. Lesson 5-R
Review of Chapter 5

2. Objectives
Find the area under the curve using Riemann sums (using either left, right, midpoint rectangular or trapezoidal approximations)
Find Anti-derivatives (differentiation in reverse)
Solve indefinite integrals with initial conditions (position, velocity and acceleration problems)
Apply Fundamental Theorem of Calculus
Solve definite integrals
Apply Total distance versus displacement
Use u-substitution to solve integrals

3. Vocabulary
None new

4. Chapter 5 Test 38 Total Problems (including 5 bonus) Riemann Sums
Total Distance versus Displacement
Anti-differentiation Rules
Indefinite Integration
U-Substitutions
Definite Integration
Fundamental Theorem of Calculus

5. Review Session Problems
Estimate the area between the curve 4 – x2 and the x-axis using 4 sub-intervals.
Using LH Rectangles:
Using Trapezoids:
Find the exact area:
Area = ht ∙ width
Area = ½ (ht1 + ht2) ∙ width
t =2 ∫
(4 - x2) dx = 4x - 1/3t3 = (8 – 8/3) = 5.33
t = 0

6. Review Session Problems
Find the displacement of an object between t = 0 and t = 3 seconds whose velocity is given by v(t) = t2 – 4 (ft/sec)
t =3
t =3 | ∫
(t2 – 4) dt = 1/3 t3 – 4t = (9 – 12) = -3 feet
t = 0
t = 0
Find the total distance traveled of the same object in the same time period.
t =2
t =3 | | ∫ ∫
(t2 – 4) dt (t2 – 4) dt =
t = 0
t = 2

7. Review Session Problems

8. Summary & Homework Summary: Homework:
Area under the curve is an integral application and can be approximated by Riemann Sums
To solve integrals, we need to know derivatives
Integration is Anti-differentiation
FTC has two parts (both very important)
Total Area is always positive
U-Substitution can help get some integrals into a form we can integrate
Homework:
Study for Chapter 5 Test