Simpson's Rule is a numerical integration technique that provides a highly accurate approximation of definite integrals, especially for smooth functions. It uses parabolic arcs to approximate the area under a curve, requiring an even number of subintervals. The rule's formula involves specific coefficients and the subinterval width, and its accuracy surpasses the Trapezoidal Rule. Understanding its theoretical basis, practical application, and error assessment is crucial for students and professionals in fields requiring numerical analysis.
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Simpson's Rule is a method for approximating the definite integral of a function when it cannot be computed analytically
Fitting parabolic arcs to the curve
Simpson's Rule offers a more accurate approximation than the Trapezoidal Rule by fitting parabolic arcs to the curve being integrated
Dividing the area under the curve into segments
Simpson's Rule involves dividing the area under the curve into segments and approximating each by a quadratic polynomial
Simpson's Rule is grounded in the concept that any three non-collinear points can be used to define a unique quadratic polynomial
When applying Simpson's Rule, the interval of integration is split into an even number of subintervals
Fitting a quadratic polynomial to each pair of subintervals
A quadratic polynomial is fitted to the function over each pair of subintervals when applying Simpson's Rule
Calculating the area under the curve for each segment
The area under the curve for each segment is calculated by integrating the corresponding quadratic polynomial
The formula for Simpson's Rule is an algebraic expression that approximates the integral of a function over a closed interval
The error in Simpson's Rule can be quantified using both relative and absolute error measures
An error bound can be calculated using the function's fourth derivative, providing an upper limit on the potential error
Simpson's Rule is highly accurate for polynomial functions of degree three or lower, but has constraints such as the necessity for an even number of subintervals and potential limitations for functions with significant oscillations
Simpson's Rule can be applied to estimate the integral of a function over a certain interval by calculating the subinterval width and summing the contributions from each subinterval
The error bound formula can be used to adjust the number of subintervals and meet a specific accuracy requirement
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When calculating the definite integral of a function, Simpson's Rule approximates the area by using ______ arcs instead of straight lines.
parabolic
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Simpson's Rule Interval Division
Divides integration interval into even number of subintervals for applying quadratic fits.
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Simpson's Rule Area Calculation
Calculates area under each segment by integrating the fitted quadratic polynomial.
