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- 2.5: Numerical Integration - Midpoint, Trapezoid, Simpson's rule
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The antiderivatives of many functions either cannot be expressed or cannot be expressed easily in closed form that is, in terms of known functions.

Consequently, rather than evaluate definite integrals of these functions directly, we resort to various techniques of numerical integration to approximate their values. In this section we explore several of these techniques.

In addition, we examine the process of estimating the error in using these techniques. Earlier in this text we defined the definite integral of a function over an interval as the limit of Riemann sums. Formally, we state a theorem regarding the convergence of the midpoint rule as follows. Compare the result with the actual value of this integral. We can also approximate the value of a definite integral by using trapezoids rather than rectangles.

The trapezoidal rule for estimating definite integrals uses trapezoids rather than rectangles to approximate the area under a curve. The areas of the remaining three trapezoids are. Generalizing, we formally state the following rule.

First of all, it is useful to note that. The trapezoidal rule tends to overestimate the value of a definite integral systematically over intervals where the function is concave up and to underestimate the value of a definite integral systematically over intervals where the function is concave down. On the other hand, the midpoint rule tends to average out these errors somewhat by partially overestimating and partially underestimating the value of the definite integral over these same types of intervals.

This leads us to hypothesize that, in general, the midpoint rule tends to be more accurate than the trapezoidal rule. An important aspect of using these numerical approximation rules consists of calculating the error in using them for estimating the value of a definite integral. We first need to define absolute error and relative error. Definition: absolute and relative error. Use the previous examples as a guide. In the two previous examples, we were able to compare our estimate of an integral with the actual value of the integral; however, we do not typically have this luxury.

In general, if we are approximating an integral, we are doing so because we cannot compute the exact value of the integral itself easily. Therefore, it is often helpful to be able to determine an upper bound for the error in an approximation of an integral. The following theorem provides error bounds for the midpoint and trapezoidal rules. The theorem is stated without proof.

Error Bounds for the Midpoint and Trapezoidal Rules. From the error-bound Equation, we have. We need to keep in mind that the error estimates provide an upper bound only for the error. The actual estimate may, in fact, be a much better approximation than is indicated by the error bound. With the midpoint rule, we estimated areas of regions under curves by using rectangles. In a sense, we approximated the curve with piecewise constant functions.

With the trapezoidal rule, we approximated the curve by using piecewise linear functions. What if we were, instead, to approximate a curve using piecewise quadratic functions?

We partition the interval into an even number of subintervals, each of equal width. As we go through the derivation, we need to keep in mind the following relationships:.

Combining these two approximations, we get. The pattern continues as we add pairs of subintervals to our approximation. The general rule may be stated as follows.

The bound in the error is given by the following rule:. After substituting, we have. The Midpoint Rule Earlier in this text we defined the definite integral of a function over an interval as the limit of Riemann sums. The Trapezoidal Rule We can also approximate the value of a definite integral by using trapezoids rather than rectangles.

Absolute and Relative Error An important aspect of using these numerical approximation rules consists of calculating the error in using them for estimating the value of a definite integral.

Hint Use the previous examples as a guide. Error Bounds on the Midpoint and Trapezoidal Rules In the two previous examples, we were able to compare our estimate of an integral with the actual value of the integral; however, we do not typically have this luxury.

Key Concepts We can use numerical integration to estimate the values of definite integrals when a closed form of the integral is difficult to find or when an approximate value only of the definite integral is needed. The midpoint rule approximates the definite integral using rectangular regions whereas the trapezoidal rule approximates the definite integral using trapezoidal approximations. Notes added to development of area under a parabola and typos fixed in original text.

The antiderivatives of many functions either cannot be expressed or cannot be expressed easily in closed form that is, in terms of known functions. Consequently, rather than evaluate definite integrals of these functions directly, we resort to various techniques of numerical integration to approximate their values. In this section we explore several of these techniques. In addition, we examine the process of estimating the error in using these techniques. Earlier in this text we defined the definite integral of a function over an interval as the limit of Riemann sums. Formally, we state a theorem regarding the convergence of the midpoint rule as follows.

trapezoidal rule questions and answers pdf How many evaluations of the function f(x) are needed in this case? (Give the least possible num- ber). This method is.

In mathematics , and more specifically in numerical analysis , the trapezoidal rule also known as the trapezoid rule or trapezium rule —see Trapezoid for more information on terminology is a technique for approximating the definite integral. It follows that. The trapezoidal rule may be viewed as the result obtained by averaging the left and right Riemann sums , and is sometimes defined this way. The integral can be even better approximated by partitioning the integration interval , applying the trapezoidal rule to each subinterval, and summing the results.

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Another useful integration rule is the Trapezoidal Rule. Under this rule, the area under a curve is evaluated by dividing the total area into little trapezoids rather than rectangles. Substituting the values of the function from the table, we find the approximate area under the curve:. The function values are known from the table, so we can easily calculate the approximate value of the area:. Necessary cookies are absolutely essential for the website to function properly. This category only includes cookies that ensures basic functionalities and security features of the website. These cookies do not store any personal information.

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The antiderivatives of many functions either cannot be expressed or cannot be expressed easily in closed form that is, in terms of known functions. Consequently, rather than evaluate definite integrals of these functions directly, we resort to various techniques of numerical integration to approximate their values. In this section, we explore several of these techniques.

*The method is of great generality and mathematical simplicity quot. A graph drawn in a plane in such a way that if the vertex set of the graph can be partitioned into two non empty disjoint subset X and Y in such a way that each edge of G has one end in X and one end in Y stability and convergence absolute stability. Numerical methods for nding the roots of a function The roots of a function f x are de ned as the values for which the value of the function becomes equal to zero.*

Summary Many applications of calculus involve definite integrals. If it is possible to find an antiderivative for the integrand, then the integral can be evaluated using the Fundamental Theorem. When an antiderivative is not apparent, numerical approximate methods are invoked.

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