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Double and triple integrals
Published in John Bird, Bird's Higher Engineering Mathematics, 2021
Double and triple integrals have engineering applications in finding areas, masses and forces of two-dimensional regions, and in determining volumes, average values of functions, centres of mass, moments of inertia and surface areas. A multiple integral is a type of definite integral extended to functions of more than one real variable. This chapter explains how to evaluate double and triple integrals and completes the many techniques of integral calculus explained in the preceding chapters.
Calculus
Published in Brian Vick, Applied Engineering Mathematics, 2020
Multiple integrals can be computed numerically by extending the methods for a function of one variable. Methods such as the trapezoid or Simpson’s rule can readily be applied. First, a rule is applied in one dimension with each value of the second dimension held constant. Then, the rule is applied in the second dimension to obtain a numerical integration of a double integral.
Generation of waves due to bottom disturbances in a viscous fluid
Published in Geophysical & Astrophysical Fluid Dynamics, 2022
We now need to evaluate the integral (37). It cannot be evaluated in general, but can be evaluated asymptotically for large x and t for various types of instantaneous bottom disturbances and zero initial pressure at the free surface. We note that is an even function in s, if is even in s. Further, when is even in s, the integral (37) can be written as The integrand of the above multiple integral is oscillatory in nature. This multiple integral can be approximated by the method of steepest descent (see Jeffreys and Lapwood 1957). In this method the integration path is to be oriented in such a way that the immediate neighbourhood of the saddle points produces the maximum contribution to the integral. Now, for , can be written as which is split into where , and are defined, respectively, by (41) in section 4.1, (45) in section 4.2 and (48) in section 4.3 below. In each subsection the integrals are evaluated asymptotically with the accumulated asymptotic value of η (see (51)) summarised in section 4.4.
Constrained variational problems governed by second-order Lagrangians
Published in Applicable Analysis, 2020
Consider a -class function , called multi-time second-order Lagrangian, where , , (multi-time interval in with respect to partial order product) is a hyper-parallelepiped determined by the diagonally opposite points . This second-order Lagrangian determines the following multiple integral functional Multi-time optimization problem.Find a m-sheet that extremizes (P2), among all the -class functions that satisfy the conditionsorand the following second-order PDEs (partial speed-acceleration constraints)
Assessing the effect of repair delays on a repairable system
Published in Journal of Quality Technology, 2020
Jiaxiang Cai, Candemir Cigsar, Zhi-Sheng Ye
The m-multiple integral in makes ML estimation of challenging. The curse of dimensionality is confronted when the failure number m is large. A natural idea is to treat the unobserved recurrent failure times as missing data, after which the EM framework can be invoked. Nevertheless, the E step still remains to resolve an m-dimensional integral. We examine the details of the EM algorithm in Appendix A. The study pursues direct computation of the integral. Because of the finite integration domain in model [3], some simple yet effective numerical integration techniques can be used. Depending on the maginitude of m, two methods are exploited in this study.