Trapezoid Rule Trapezoid Rule In numerical analysis, the trapezoidal rule (also known as the trapezoid rule or trapezium rule) is an approximate technique for calculating the definite integral The trapezoidal rule is one of a family of formulas for numerical integration called Newtonâ&#x20AC;&#x201C; Cotes formulas, of which the midpoint rule is similar to the trapezoid rule. Simpson's rule is another member of the same family, and in general has faster convergence than the trapezoidal rule for functions which are twice continuously differentiable, though not in all specific cases. However for various classes of rougher functions (ones with weaker smoothness conditions), the trapezoidal rule has faster convergence in general than Simpson's rule. Moreover, the trapezoidal rule tends to become extremely accurate when periodic functions are integrated over their periods, which can be analyzed in various ways. For non-periodic functions, however, methods with unequally spaced points such as Gaussian quadrature and Clenshawâ&#x20AC;&#x201C;Curtis quadrature are generally far more accurate; Clenshawâ&#x20AC;&#x201C; Curtis quadrature can be viewed as a change of variables to express arbitrary integrals in terms of periodic integrals, at which point the trapezoidal rule can be applied accurately.

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In this page, we are going to learn about trapezoidal rule concept. Below you can see the explanation about trapezoidal rule. The evaluation of a definite integral from a set of numerical values is the process of numerical integration. If the function is of a single variable, the process is called as quadrature. The numerical integration is calculated by first approximating the integral by a polynomial through an interpolation formula and then integrate it between the given limits. Thus, to evaluate ∫baf(x)dx ,express the function f(x) by an interpolation formula, say p(x) and then, integrate the function f(x) by an interpolation formula, say p(x) and then, integrate p(x) between the limits a and b. Therefore, ∫baf(x)dx ∼ ∫bap(x)dx There are times when it is not possible to evaluate a definite integral directly, using the standard method. ∫baf(x)dx=[F(x)]ba=F(b)–F(a), where I(x) is the simplest function for which ddxF(x)=f(x). Two examples which you cannot integrate with your knowledge so far are ∫1011+x2dx and ∫101+x2−−−−−√dx. You need to use another method for approximating to the integrals. Divide the interval from a to b into n equal intervals, each of width h, so that h = b−an Call the x-coordinate of the left side of the first Interval x0, so x0 = a, and then successively let x1=x0+h,x2=x0+2h and so on until xn−1=x0+(n−1)h and xn−x0+nh=b.

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To shorten the amount of writing, use the shorthand y0=f(x0),y1=f(x1) and so on. Then, using the simple form of the each interval of width h in turn, you find that ∫baf(x)dx = 12h(y0+y1)+12h(y1+y2)+12h(y2+y3)+......+12h(yn−1+yn) = 12h[(y0+yn)+2(y1+y2+....+yn−1)] The sum estimates total area under the curve y = f(x) on the interval a and b and hence also estimates the integral ∫baf(x)dx. This approximation formula is known as the trapezoid rule and applies as a mean of approximating the integral even if the function is not positive. Therefore, the trapezoidal rule with n intervals states that ∫baydx=12h[(y0+yn)+2(y1+y2+....+yn−1)] where h = b−an Now, let us try some problems to apply the trapezoidal rule.

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Trapezoid Rule