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25 lip 2021 · The most commonly used techniques for numerical integration are the midpoint rule, trapezoidal rule, and Simpson’s rule. The midpoint rule approximates the definite integral using rectangular regions whereas the trapezoidal rule approximates the definite integral using trapezoidal approximations.
The Composite Midpoint Rule. Using the Midpoint Rule on each interval and summing gives a formula that could be familiar: M n := f ( x 0 + x 1 2) h + f ( x 1 + x 2 2) h + ⋯ + f ( x n − 1 + x n 2) h = f ( a + ( a + h) 2) h + f ( ( a + h) + ( a + 2 h) 2) h + ⋯ + f ( ( b − h) + b 2) h = [ f ( a + h / 2) + f ( a + 3 h / 2) + ⋯ + f ( b − h / 2)] h.
Numerical Integration: Composite Midpoint Rule Midpoint Rule (1-point open Newton-Cotes formula) f(x) dx = where < < Theorem: Composite Midpoint Rule Let f e C2[a, b], n be even, h = (b — a)/(n + 2), and —a + (j + l)h n + 1. There exists a [L e (a, b) for which the for each j 10
Spring 2020. The point: Techniques for computing integrals are derived, using interpolation and piece-wise constructions (composite formulas). In addition, the asymptotic error series for the trapezoidal rule is introduced, enabling the use of Richardson extrapolation for integration.
•Example: Use both Composite Simpson’s rule and Composite Trapezoidal rule to approximate with an absolute error less than 0.00002. ∫π 0 sin xdx •
Example 1. a) Use Simpson’s rule to approximate ∫𝑒𝑒𝑥𝑥𝑑𝑑𝑑𝑑 4 0. The exact value is 53.59819. b) Divide [0,4] into [0,1] + [1,2] + [2,3] + [3,4]. Use Simpson’s rule to approximate ∫𝑒𝑒𝑥𝑥𝑑𝑑𝑑𝑑 1 0, ∫𝑒𝑒𝑥𝑥𝑑𝑑𝑑𝑑 2 1, ∫𝑒𝑒𝑥𝑥𝑑𝑑𝑑𝑑 3 2
The midpoint rule approximates the area between the graph of f(x) f ( x) and the x-axis by summing the areas of rectangles with midpoints that are points on f(x) f ( x). Example: Using the Midpoint Rule with M 4 M 4. Use the midpoint rule to estimate ∫ 1 0 x2dx ∫ 0 1 x 2 d x using four subintervals.
