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1. The trapezoidal rule is mostly used in the numerical analysis process.
2. The Trapezoidal Rule does not give accurate value as Simpson’s Rule when the underlying function is smooth.
3. and x=8 using Trapezoidal Rule with n = 4 subintervals.
4. using Trapezoidal Rule with n = 6 subintervals.
5. Example 7 Approximate the integral \(\int\limits_0^1 {{x^3}dx}\) using the Trapezoidal Rule with \(n = 2\) subintervals.
6. Example 8 Approximate the integral \(\int\limits_0^2 {{x^2}dx}\) using the Trapezoidal Rule with \(n = 3\) subintervals.
7. The trapezoidal rule for estimating definite integrals uses trapezoids rather than rectangles to approximate the area under a curve.
8. Before continuing, let’s make a few observations about the trapezoidal rule.
9. This leads us to hypothesize that, in general, the midpoint rule tends to be more accurate than the trapezoidal rule.
10. With the trapezoidal rule, we approximated the curve by using piecewise linear functions.
11. For the implicit trapezoidal rule for solving initial value problems, see Trapezoidal rule (differential equations) .
12. The trapezoidal rule may be viewed as the result obtained by averaging the left and right Riemann sums, and is sometimes defined this way.
13. trapezoidal rule is usually what is meant by "integrating with the trapezoidal rule".
14. The trapezoidal rule converges rapidly for periodic functions.
15. In the case of \(\alpha =-1\), this approach is the well known trapezoidal rule for numerical integration.
16. hence, the Trapezoidal rule gives exact value of the integral if the integrand is a linear function.
17. In Section 2, after introducing some basic formulas of the general (composite) trapezoidal rule and notations, we present our main result.
18. The trapezoidal rule is one method we can use to approximate the area under a function over a given interval.
19. If it’s difficult to find area exactly using an integral, we can use trapezoidal rule instead to estimate the integral.
20. The idea of the trapezoidal rule is to approximate a general curve by trapezoids, like this.
21. However, for very jagged functions, the trapezoidal rule can be more accurate.
22. The trapezoidal rule has often been referred to as being symmetric or time-reversible and is therefore good for Hamiltonian systems.
23. In this paper, we show that the trapezoidal rule preserves a symplectic structure different from the original one by O ( h 2 ).
24. The ideas in this paper also motivate us to apply Richardson's extrapolation to the trapezoidal rule.
25. Can I Use Trapezoidal Rule to Calculate an Improper Integral?
26. Trapezoidal Rule is mostly used for evaluating the area under the curves.

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• [{'LOWER': 'trapezoidal'}, {'LEMMA': 'rule'}]