 Georg Friedrich Bernhard Riemann

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

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
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
5.2 The Definite Integral

Properties Continued 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