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

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. So, the limit of sum is

Definition of a Riemann Sum

The Definite Integral = length of the i th subinterval a b = partition of [a, b] = length of the i th subinterval a Norm of b = length of the longest subinterval Upper Limit definition area, f(x) > 0 on [a, b] Lower Limit net area, otherwise Riemann Sum - approximates the definite integral “definite integral of f from a to b”

variable of integration upper limit of integration Integration Symbol integrand variable of integration (dummy variable) lower limit of integration 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 Is 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?

About Theorem 4.4 Continuity Implies Integrability Answers False. Counterexample is Yes. 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

The Definite Integral f A a b A1 f A3 = area above – area below a b A2

Special Cases If using subintervals of equal length, (regular partition), with ci 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 ci chosen as the left endpoint of the i th subinterval, then Regular Left-Endpoint Formula (RL-EF)

Theorem 4.6 Properties of the Definite Integral c b by definition by definition

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.

Theorem 4.7 Properties of the Definite Integral

Examples Example 2 If and then find Solution

Homework Pg. 278 9, 13-19 odd, 25-31 odd, 33-41 odd, 45-49, 55