Calculus II Numerical Integration Project Fall 2016.docx

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CALCULUS II NUMERICAL INTEGRATION PROJECT FALL 2016 Course: MATH 2212 Name: HUY TRUONG Professor: SHARON L. SANDERS Due Day : OCTOBER 19 TH
Calculus II Numerical Integration Project Fall 2016 Introduction Purpose : Sometimes it is necessary to compute a definite integral that cannot be handled by formulas or integration techniques. In this case, the definite integral can be approximated. There are several numerical methods for this approximation: the rectangle method (studied in Calculus I), the Trapezoidal Rule, Simpson’s Rule, or the graphing calculator(function integrate feature). These methods also have applications to real world data tables and graphs representing rates of change, areas and volumes. The Trapezoidal Rule uses trapezoids instead of rectangles to approximate the area under a curve above the x axis (a definite integral). Simpson’s Rule uses parabolas to approximate the area under a curve above the x-axis (a definite integral). In both cases, the interval of integration from a to b is divided equally into n parts so that there are n partition points in each interval. Trapezoidal Rule: f ( x 0 ) + 2 f ( ¿ x 1 )+ 2 f ( x 2 ) + 2 f ( x 3 ) + + 2 f ( x n 1 ) + f ( x n ) f ( x ) dx = 1 2 b a n ¿ a b ¿ ] Where n represents the number of trapezoids used for the approximation. Accuracy increases as n increases (the more trapezoids, the better the result). The number n will be given in each problem. Simpson’s Rule: a b f ( x ) dx = 1 3 b a n [ f ( x 0 ) + 4 f ( x 1 ) + 2 f ( x 2 ) + 4 f ( x 3 ) + 2 f ( x 4 ) + ... + 4 f ( x n 1 ) + f ( x n ) ] Simpson’s Rule is more accurate than the Trapezoidal Rule for the same value of n. Accuracy increases as n increases. The n will be given in each problem. The n must be an even number.

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