Georg Friedrich Bernhard Riemann Chapter 5 – Integrals 5.2 The Definite Integral 5.2 The Definite Integral Georg Friedrich Bernhard Riemann 1826 - 1866

Review - Riemann Sum When we find the area under a curve by adding rectangles, the answer is called a Riemann sum. The width of a rectangle is called a subinterval. The entire interval is called the partition. subinterval partition Subintervals do not all have to be the same size. 5.2 The Definite Integral

Example 1 – pg. 382 # 4 If you are given the information above, evaluate the Riemann sum with n=6, taking sample points to be right endpoints. What does the Riemann sum illustrate? Illustrate with a diagram. Repeat part a with midpoints as sample points. 5.2 The Definite Integral

Idea of the Definite Integral is called the definite integral of over . If we use subintervals of equal length, then the length of a subinterval is: The definite integral is then given by: 5.2 The Definite Integral

Definite Integral in Leibnitz Notation Leibnitz introduced a simpler notation for the definite integral: Note that the very small change in x becomes dx. 5.2 The Definite Integral

Explanation of the Notation 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. 5.2 The Definite Integral

Theorem (3) If f is continuous on [a, b], or if f has only a finite number of jump discontinuities, then f is integrable on [a, b]; that is, the definite integral exists. 5.2 The Definite Integral

Theorem (4) Putting all of the ideas together, if f is differentiable on [a, b], then where 5.2 The Definite Integral

Example 2 Use the midpoint rule with the given value of n to approximate the integral. Round your answers to four decimal places. 5.2 The Definite Integral

Evaluating Integrals using Sums 1. 2. 3. 4. 5. 6. 7. 5.2 The Definite Integral

Example 4 – Page 377 #23 Use the form of the definition of the integral given in Theorem 4 to evaluate the integral. 5.2 The Definite Integral

Example 5 Use the form of the definition of the integral given in Theorem 4 to evaluate the integral. 5.2 The Definite Integral

Example 6 – Page 383 # 29 Express the integral as a limit of Riemann sums. Do not evaluate the limit. 5.2 The Definite Integral

Example 7 – page 383 # 17 Express the limit as a definite integral on the given interval. 5.2 The Definite Integral

Example 8 – page 385 # 71 Express the limit as a definite integral. 5.2 The Definite Integral

Properties of the Integral 1. 2. 3. 4. 5.2 The Definite Integral

Properties Continued 5. 6. 7. 8. 5.2 The Definite Integral

Example 9 – page 384 # 62 Use Property 8 to estimate the value of the integral. 5.2 The Definite Integral

Example 10 – page 384 # 37 Evaluate the integral by interpreting it in terms of areas. 5.2 The Definite Integral

Example 11 – page 383 # 28 Work in groups to prove the following: 5.2 The Definite Integral

What to expect next… We will be evaluating Leibnitz integrals using the idea of antiderivatives and the fundamental theorem of calculus. 5.2 The Definite Integral