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The Midpoint Rule is a numerical method used to approximate the value of a definite integral. It provides a way to estimate the area under a curve, which is particularly useful when the integral cannot be calculated directly.
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.
Midpoint rule — Numerical_Analysis. 5.4. Midpoint rule #. Assume that { x 0, x 1, …, x n } are n + 1 in [ a, b] such that. and Δ x j is defined as Δ x j = x j + 1 − x j. Then, where x j ∗ = ( x j + x j + 1) / 2, for 0 ≤ j ≤ n − 1 are the midpoint of the intervals.
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.
Use the midpoint rule to estimate [latex]{\displaystyle\int }_{0}^{1}{x}^{2}dx[/latex] using four subintervals. Compare the result with the actual value of this integral.
Presentation on theme: "Midpoint and Trapezoidal Rules"— Presentation transcript: 1 Midpoint and Trapezoidal Rules 2 Approximating Integrals
Gaussian Quadrature. The quadrature rule is defined by interpolation points xi 2 [a; b], x1 < x2 < < xn; and weights wi to multiply the function values with. The cost of a quadrature rule is determined by the number of function values, or equivalently, the number of interpolation points.
