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Riemann Sums Definite Integrals

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Riemann Sums Definite Integrals

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    1. Riemann Sums & Definite Integrals Section 5.3

    2. Finding Area with Riemann Sums For convenience, the area of a partition is often divided into subintervals with equal width in other words, the rectangles all have the same width. (see the diagram to the right and section 5-2)

    3. Finding Area with Riemann Sums It is possible to divide a region into different sized rectangles based on an algorithm or rule (see graph above and example #1 on page 307)

    4. Finding Area with Riemann Sums It is also possible to make rectangles of whatever width you want where the width and/or places where to take the height does not follow any particular pattern.

    5. What is a Riemann Sum Definition: Let f be defined on the closed interval [a,b], and let ? be a partition [a,b] given by a = x0 < x1 < x2 < < xn-1 < xn = b where ?xi is the width of the ith subinterval [xi-1, xi]. If c is any point in the ith subinterval, then the sum

    6. New Notation for ?x When the partitions (boundaries that tell you where to find the area) are divided into subintervals with different widths, the width of the largest subinterval of a partition is the NORM of the partition and is denoted by ||?|| If every subinterval is of equal width, the partition is REGULAR and the norm is denoted by ||?||= ?x = The number of subintervals in a partition approaches infinity as the norm of the partition approaches 0. In other words, ||?|| 0 implies that n ?

    7. Definite Integrals If f is defined on the closed interval [a,b] and the limit

    8. Definite Integrals vs. Indefinite Integrals A definite integral is number. An indefinite integral is a family of functions. They may look a lot alike, however, definite integrals have limits of integration while the indefinite integrals have not limits of integration.

    9. Theorem 5.4 Continuity Implies Integrability If a function f is continuous on the closed interval [a,b], then f is integrable on [a,b].

    10. Evaluating a definite integral To learn how to evaluate a definite integral as a limit, study Example #2 on p. 310.

    11. Theorem 5.5 The Definite Integral as the Area of a Region If f is continuous and nonnegative on the closed interval [a,b], then the area of the region bounded by the graph of f , the x-axis, and the vertical lines x = a and x = b is given by

    12. Lets try this out Sketch the region Find the area indicated by the integral.

    13. Give this one a try Sketch the region Find the area indicated by the integral.

    14. Try this one Sketch the region Find the area indicated by the integral.

    15. Properties of Definite Integrals If f is defined at x = a, then we define

    16. Additive Interval Property

    17. Properties of Definite Integrals

    18. HOMEWORK yepmore practice Thursday, January 17 Read and take notes on section 5.3 and do p. 314 # 3, 6, 9, , 45, 47, 49, 53, 65- 70 Work on the AP Review Diagnostic Tests if you have time over the long weekend. Tuesday January 22 Get the Riemann sum program for your calculator and do p. 316 # 59-64, 71 and then READ and TAKE Notes on section 5-4 and maybe more to come!!

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