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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

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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

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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

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**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

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**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

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**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

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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

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

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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

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**Evaluating Integrals using Sums**

5.2 The Definite Integral

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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

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

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

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

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**Example 8 – page 385 # 71 Express the limit as a definite integral.**

5.2 The Definite Integral

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**Properties of the Integral**

5.2 The Definite Integral

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Properties Continued 5.2 The Definite Integral

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

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

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**Example 11 – page 383 # 28 Work in groups to prove the following:**

5.2 The Definite Integral

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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

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