Trapezoidal Rule
(b-a/2n) [f(xo + 2f(x1)+…+2f(xn-1)+f(xn)]
The trapezoidal rule is a numerical integration method used to approximate definite integrals of a function. It involves approximating the area under the curve of the function using trapezoids.
The formula for the trapezoidal rule is:
∫a^b f(x)dx ≈ (b-a) * [f(a) + f(b)] / 2
where a and b are the lower and upper limits of integration, f(x) is the function being integrated over that interval, and the fractional approximation is the area of the trapezoid with base lengths equal to a and b and a height equal to the average value of f(a) and f(b).
In other words, the trapezoidal rule divides the area under the curve into trapezoids and adds up the areas of the trapezoids to approximate the area under the curve.
The accuracy of the trapezoidal rule depends on the number of trapezoids used to approximate the integral. Generally, the more trapezoids used, the more accurate the approximation will be. However, using too many trapezoids can lead to numerical instability and rounding errors.
The trapezoidal rule is often used as a simple and quick method to approximate integrals, especially when the exact integral is difficult to calculate analytically. It is also commonly used as a starting point for more advanced numerical integration methods.
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