Numerical integration
Approximating a definite integral with rectangles, trapezoids and Simpson's rule: composite formulas, the sign of the error according to monotonicity and convexity, and the order of each method.
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Numerical integration: what is it?
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Computing an approximate value of an integral by replacing the function with simple pieces whose area is known.
Useful when the function has no closed-form antiderivative (e^(−x²)) or is only known through measurements; the aim is the best trade-off between the number of evaluations of f and the error.
We approximate the integral of f(x) dx from 2 to 5 with n = 6 subintervals. What is the step h?
- 0.5
- 3
- 2
- 18
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0.5
h = (b − a)/n = (5 − 2)/6 = 0.5. The value 3 is the length of the interval, 2 comes from n/(b − a), and 18 from (b − a) × n.
A composite method on [a; b] with n subintervals has step …; the trapezoidal rule evaluates f at … points.
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A composite method on [a; b] with n subintervals has step h = (b − a)/n; the trapezoidal rule evaluates f at n + 1 points.
n subintervals have n + 1 endpoints, as n fence sections have n + 1 posts. Each evaluation of f has a cost: that is what we try to save.
Definite integral: what is it?
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The signed area between the curve of f, the x-axis and the lines x = a and x = b, counted negative below the axis.
It is the limit of the sums of rectangles as the step tends to 0: it is this definition that numerical methods approximate with a finite step.