2 The Definite IntegralIn the Section 4.2, the definition of area is defined as
3 The Definite IntegralThe following example shows that it is not necessary to have subintervals of equal widthExample Find the area bounded by the graph ofand x-axis over the interval [0, 1].SolutionLet ( i = 1, 2, …, n) be the endpoint of the subinteravls. Then the width of the i th subinterval isThe width of all subintervals varies.Let ( i = 1, 2, …, n) be the point in the i th subinteravls, then
4 Continued… Example 1 Find the area bounded by the graph of and x-axis over the interval [0, 1].SolutionLet and ( i = 1, 2, …, n) be the endpoint of the subinteravls and the point in the i th subinterval.So, the limit of sum is
6 The Definite Integral = length of the i th subinterval a b = partition of [a, b]= length of the i th subintervalaNorm ofb= length of the longest subintervalUpper Limitdefinitionarea, f(x) > 0 on [a, b]Lower Limitnet area, otherwiseRiemann Sum - approximates the definite integral“definite integral of f from a to b”
7 variable of integration upper limit of integrationIntegrationSymbolintegrandvariable of integration(dummy variable)lower limit of integrationIt is called a dummy variable because the answer does not depend on the variable chosen.
9 Theorem 4.4 Continuity Implies Integrability QuestionsIs the converse of Theorem 4.4 true? Why?If change the condition of Theorem 4.4 “f is continuous” to “f is differentiable”, is the Theorem 4.4 true?Of the conditions “continuity”, “differentiability” and “integrability”, which one is the strongest?
10 About Theorem 4.4 Continuity Implies Integrability AnswersFalse. Counterexample isYes. Because “f is differentiable” implies “f is continuous”The order from strongest to weakest is “integrability”, “continuity”, and “differentiability”.1, when x ≠ 1 on [0, 5]0, otherwise
11 The Definite IntegralfAabA1fA3= area above – area belowabA2
12 Special CasesIf using subintervals of equal length, (regular partition), with ci chosen as the right endpoint of the i th subinterval, thenRegular Right-Endpoint Formula (RR-EF)
13 Special CasesIf using subintervals of equal length, (regular partition), with ci chosen as the left endpoint of the i th subinterval, thenRegular Left-Endpoint Formula (RL-EF)
14 Theorem 4.6 Properties of the Definite Integral cbby definitionby definition
15 Theorem 4.6 Properties of the Definite Integral 1.Reversing the limits changes the sign.2.If the upper and lower limits are equal, then the integral is zero.3.Constant multiples can be moved outside.4.Integrals can be added and subtracted.
16 Theorem 4.7 Properties of the Definite Integral