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CHAPTER 4 SECTION 4.3 RIEMANN SUMS AND DEFINITE INTEGRALS

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Riemann Sum 1.Partition the interval [a,b] into n subintervals a = x 0 < x 1 … < x n-1 < x n = b Call this partition P The k th subinterval is x k = x k-1 – x k Largest x k is called the norm, called || || If all subintervals are of equal length, the norm is called regular. 2.Choose an arbitrary value from each subinterval, call it

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Riemann Sum 3.Form the sum This is the Riemann sum associated with the function f the given partition P the chosen subinterval representatives We will express a variety of quantities in terms of the Riemann sum

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This illustrates that the size of ∆x is allowed to vary a x 1 x 2 x 3 x 4 x 5 x 1 * x 2 * x 3 * x 4 * x 5 * Then a < x 1 < x 2 < x 3 < x 4 ….etc. is a partition of [ a, b ] Notice the partition ∆x does not have to be the same size for each rectangle. y = f (x) And x 1 *, x 2 *, x 3 *, etc… are x coordinates such that a < x 1 * < x 1, x 1 < x 2 * < x 2, x 2 < x 3 * < x 3, … and are used to construct the height of the rectangles. Etc…

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c k. The graph of a typical continuous function y = ƒ(x) over [a, b]. Partition [a, b] into n subintervals a < x 1 < x 2 <…x n < b. Select any number in each subinterval c k. Form the product f(c k ) x k. Then take the sum of these products.

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Riemann Sum This is called the Riemann Sum of the partition of x. norm The width of the largest subinterval of a partition is the norm of the partition, written ||x||. As the number of partitions, n, gets larger and larger, the norm gets smaller and smaller. As n , ||x|| 0 only if ||x|| are the same width!!!!

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The Riemann Sum Calculated Consider the function 2x 2 – 7x + 5 Use x = 0.1 Let the = left edge of each subinterval Note the sum

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The Riemann Sum We have summed a series of boxes If the x were smaller, we would have gotten a better approximation f(x) = 2x 2 – 7x + 5

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Finer partitions of [a, b] create more rectangles with shorter bases.

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The Definite Integral The definite integral is the limit of the Riemann sum We say that f is integrable when –the number I can be approximated as accurate as needed by making || || sufficiently small –f must exist on [a,b] and the Riemann sum must exist – is the same as saying n

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Integration Symbol lower limit of integration upper limit of integration integrand variable of integration Notation for the definite integral

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Important for AP test [ and mine too !! ] Recognizing a Riemann Sum as a Definite integral

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From our textbook Notice the text uses ∆ instead of ∆x, but it is basically the same as our ∆x, and c i is our x i *

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Try the reverse : write the integral as a Riemann Sum … also on AP and my test

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Theorem 4.4 Continuity Implies Integrability

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DC I Relationship between Differentiability, Continuity, and Integrability D – differentiable functions, strongest condition … all Diff ’ble functions are continuous and integrable. C – continuous functions, all cont functions are integrable, but not all are diff ’ble. I – integrable functions, weakest condition … it is possible they are not con‘ t, and not diff ‘ble.

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Evaluate the following Definite Integral First … remember these sums and definitions: c i = a + i x

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Evaluate the definite integral by the limit definition EXAMPLE Evaluate the definite integral by the limit definition

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Evaluate the definite integral by the limit definition, continued

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The Definite integral above represents the Area of the region under the curve y = f ( x), bounded by the x-axis, and the vertical lines x = a, and x = b y = f ( x ) ab y x

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Theorem 4.4 Continuity Implies Integrability

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DC I Relationship between Differentiability, Continuity, and Integrability D – differentiable functions, strongest condition … all Diff ’ble functions are continuous and integrable. C – continuous functions, all cont functions are integrable, but not all are diff ’ble. I – integrable functions, weakest condition … it is possible they are not con‘ t, and not diff ‘ble.

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3 Y = x x y Areas of common geometric shapes Sol’n to definite integral A = ½ base * height 0

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A Sight Integral... An integral you should know on sight This is the Area of a semi-circle of radius a a-a

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Special Definite Integrals for f (x ) integrable from a to b

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EXAMPLE

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Additive property of integrals ab c y x

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More Properties of Integrals

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EXAMPLE

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Even – Odd Property of Integrals Even function: f ( x ) = f ( - x ) … symmetric about y - axis

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Finally …. Inequality Properties END

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Rules for definite integrals Evaluate the using the following values: Example 2: = 60 + 2(2) = 64

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check Using the TI 83/84 to check your answers Find the area under on [1,5] Graph f(x) Press 2 nd CALC 7 Enter lower limit 1 Press ENTER Enter upper limit 5 Press ENTER.

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Set up a Definite Integral for finding the area of the shaded region. Then use geometry to find the area.

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Use the limit definition to find

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Set up a Definite Integral for finding the area of the shaded region. Then use geometry to find the area. rectangle triangle

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