Lesson 5-R Review of Chapter 5

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

Vocabulary None new

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

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

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) - 0 = -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

Review Session Problems

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