RAMs.notebook December 04, 2017

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1 RAMsnotebook December 04, 2017 Riemann Sums and Definite Integrals Estimate the shaded area Area between a curve and the x-axis How can you improve your estimate? Suppose f(x) 0 x [a, b], then the area between the curve and the x-axis can be approximated by using the sum of the areas of many rectangles with bases on the x-axis and heights on the function height = f(x i ) Choosing width and height of the rectangles: Widths of each sub-interval can vary The way the interval is divided into subintervals is called a "partition", denoted P The subinterval with the longest width is the "norm" of the partition and is denoted P Ex: if [2, 10] is subdivided by {2, 3, 5, 6, 75, 10} width = Δx Height of the rectangle will be a point on the function somewhere in the closed subinterval height = f(x i ) where x i is any point in the interval Estimate the area using the intervals shown, with the height at the given point in the interval

2 RAMsnotebook December 04, 2017 Recall: "k" is the index of summation "i" is the lower limit and is the first integer substituted in for k "n" is the upper limit and is the last integer substituted in for k Summation notation for the sum of the areas of a positive function f: Let the width of a rectangle be Δx k, and let w k be some value within the k th interval so that the height of a rectangle is f(w k ), and let there be n subintervals, Then A f(w2) Δx 2 The exact area will be the limit of the sums of the area of the rectangles as the number of subintervals approaches infinity Rectangular Approximation Methods: RAMs Simplifies the choosing process Alternately, use the sum of the areas as the norm of the partitions approaches 0 Note that this is only for a function f that is non-negative in each interval, If a function has negative values then use the absolute value Partitions are often "regular", that is, each subinterval has the same width Determined by the number of subintervals (rectangles) you want Equal widths from x = a to x = b with n sub intervals will be b - a Δx = n Good choices for the height are: Right (RRAM), Left(LRAM), Middle(MRAM), Upper (maximum or circumscribed) or Lower (minimum or inscribed) Ex: If [2, 10] is to have a regular partition of 16 subintervals what is P?

3 RAMsnotebook December 04, 2017 LRAM RRAM MRAM Upper (maximum or circumscribed) Lower (minimum or inscribed) *RAM = Rectangular Approximation Method Let f(x) = x² + 1 on the interval [0, 2] and let n = 4, be the number of equal subintervals 1 Draw the RRAM, LRAM, MRAM, and find the approximate area using those rectangles Let f(x) = x² - 6x + 10 on the interval [0, 2] and let n = 4, be the number of equal subintervals 1 Draw the RRAM, LRAM, MRAM, Lower and Upper and find the approximate area using those rectangles Riemann Sums Riemann sum defines areas below the x axis as negative areas and areas above the x axis as positive areas x 0 x 1 x 2 x 5 x8 x 10 x 13 Given the partition above, the sum of the areas of each rectangle estimates the total area between the x axis and the curve The sum of the areas above the x axis minus the areas below the x axis is a Riemann Sum, [x 0, x 13]

4 RAMsnotebook December 04, 2017 The Riemann sum will be the sum of the positive and negative areas between the graph and the x-axis The exact value of the Riemann sum is a limit as P becomes negligible Alternately we can find the sum of the ± areas of the rectangles as the number of rectangles goes to infinity: This limit (provided it exists) is called the definite integral of f on the closed interval [x 0, x n ] The symbol for the definite integral of f(x) on [a, b] is where a is the lower limit, b is the upper limit, f(x) is the height of any rectangle at any x, and dx is the width of a rectangle Calculator steps: TI83: Math->9:fnInt( function,x,lower limit, upper limit) TI84: Math->9:fnInt, enter lower limit, upper limit, function and "x" after d It describes the sum of the areas above the x axis minus the sum of the areas below the x axis *Use the program provided or the website Estimate the area under the curve from x = 0 to x = 4 using 4, 10, 20 and 100 subintervals, for LRAM, RRAM and MRAM What is the best estimate of the area? 10 Suppose the areas between the graph and the x-axis are given for a function f between [a, d] 5 LRAM RRAM MRAM

5 RAMsnotebook December 04, 2017 Find LRAM, MRAM and RRAM Find LRAM and RRAM Why can't you find MRAM? Some properties, given a, b, c are constants and a < b and f and g are continuous functions, then: If f is odd then If f is even then 7 Area of f(x) in [a, b] = f(x) What is the area between the curve and the x-axis? What is the area between the x-axis and the curve?

6 RAMsnotebook December 04, 2017 Let