Presentation on theme: "4.3 Riemann Sums and Definite Integrals. The Definite Integral In the Section 4.2, the definition of area is defined as."— Presentation transcript:
4.3 Riemann Sums and Definite Integrals
The Definite Integral In the Section 4.2, the definition of area is defined as
The Definite Integral The following example shows that it is not necessary to have subintervals of equal width Example 1 Find the area bounded by the graph of and x -axis over the interval [0, 1]. Solution Let ( i = 1, 2, …, n ) be the endpoint of the subinteravls. Then the width of the i th subinterval is The width of all subintervals varies. Let ( i = 1, 2, …, n ) be the point in the i th subinteravls, then
So, the limit of sum is Continued… Example 1 Find the area bounded by the graph of and x -axis over the interval [0, 1]. Solution Let and ( i = 1, 2, …, n ) be the endpoint of the subinteravls and the point in the i th subinterval.
Definition of a Riemann Sum
= partition of [ a, b ] = length of the i th subinterval Norm of = length of the longest subinterval definite integral of f from a to b definition Riemann Sum - approximates the definite integral area, f(x) > 0 on [ a, b ] net area, otherwise The Definite Integral a b Upper Limit Lower Limit
Integration Symbol lower limit of integration upper limit of integration integrand variable of integration (dummy variable) It is called a dummy variable because the answer does not depend on the variable chosen.
Definition of a Definite Integral
Theorem 4.4 Continuity Implies Integrability Questions (1)Is the converse of Theorem 4.4 true? Why? (2)If change the condition of Theorem 4.4f is continuous tof is differentiable, is the Theorem 4.4 true? (3)Of the conditions continuity, differentiability and integrability, which one is the strongest?
Answers (1)False. Counterexample is (2)Yes. Becausef is differentiable impliesf is continuous (3)The order from strongest to weakest is integrability, continuity, and differentiability. About Theorem 4.4 Continuity Implies Integrability 1, when x 1 on [ 0, 5 ] 0, otherwise
f ab A ab f A1A1 A2A2 A3A3 = area above – area below The Definite Integral
If using subintervals of equal length, (regular partition), with c i chosen as the right endpoint of the i th subinterval, then Regular Right-Endpoint Formula (RR-EF) Special Cases
If using subintervals of equal length, (regular partition), with c i chosen as the left endpoint of the i th subinterval, then Regular Left-Endpoint Formula (RL-EF) Special Cases
f by definition a b c Theorem 4.6 Properties of the Definite Integral
2. If the upper and lower limits are equal, then the integral is zero. 1. Reversing the limits changes the sign. 3. Constant multiples can be moved outside. 4. Integrals can be added and subtracted.
Theorem 4.7 Properties of the Definite Integral