Notes 1.2.docx
Name:________________________________________Date:__________________Period:________________Calculus AB 4.2C: Definite Integrals and Numeric Integration(Note: depending on whether the function is increasing or decreasing, Ln or Rn could either be an upper or lower bound.) If we desire better approximations of the area, we could partition our area into smaller subintervals using more rectangles. The following chart shows the areas of the same region S, using n rectangles of equal width using both the left-endpoint and right-endpoint methods.The approximation of the area under the parabola f x x2 from x 0 to x1.41198803365500-170815-190500470281025717500As n approaches infinity, the area approximations approach the actual area, each converging on the true value of the area.The process of finding the sum of the areas of rectangles to approximate area of a region is called Riemann Sums, after Bernhard Riemann, who pioneered the process. Riemann proved that the finite process of adding up rectangular areas could be found by a routine analytic process know as definite integration. Here’s the essence of his great, time- saving work.ExamplesSometimes we can use known geometric formulas to come up with ACTUAL values of integrals rather than simply approximations.1905112705068570151765-115570143510Sketch the region corresponding to each definite integral. Then evaluate each integral using a geometric formula.25565108448-28575023367900479107517272000235267510604500Classwork/Homework4591553358336001) Use 4 subintervals of equal width to approximate the area under the graph fx=2x from x0 to x=4, notated as region S, using the indicated method.(a) Rectangles using the Midpoint, M4 (b) Using Trapezoids, T42) Set up a definite integral that yields the area of the region. Then evaluate each integral using a geometric formula.1333501270000(a) (b) (c)548640040830500352425038036500178117538989000190500436880003) Sketch the region corresponding to each definite integral. Then evaluate each integral using a geometric formula.(a) (b) (c) (d)5248275328295169862523630600 -47625271145335280088900000Set up a definite integral that yields the area of the region. Then evaluate each integral using a geometric formula.200442816 A=15 A=4 A=16Sketch the region corresponding to each definite integral. Then evaluate each integral using a geometric formula.-51518203476522814820347632639002032001554480203200522605015748000333375115748000-5080012573000164465030353000A=8 A=14 A= ? A= 4.5π ................
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