Proposal of an algorithmic methodology in Geo Gebra for the teaching of the Riemann sum a tool for approximating definite integrals

Abstract Background Navier-Stokes and continuity equations are utilized to simulate fully developed laminar Dean flow with an oscillating time-dependent pressure gradient. These equations are solved analytically with the appropriate boundary and initial conditions in terms of Laplace domain and inverted to time domain using a numerical inversion technique known as Riemann-Sum Approximation (RSA). The flow is assumed to be triggered by the applied circumferential pressure gradient (azimuthal pressure gradient) and the oscillating time-dependent pressure gradient. The influence of the various flow parameters on the flow formation are depicted graphically. Comparisons with previously established result has been made as a limit case when the frequency of the oscillation is taken as 0 (ω = 0). Results It was revealed that maintaining the frequency of oscillation, the velocity and skin frictions can be made increasing functions of time. An increasing frequency of the oscillating time-dependent pressure gradient and relatively a small amount of time is desirable for a decreasing velocity and skin frictions. The fluid vorticity decreases with further distance towards the outer cylinder as time passes. Conclusion Findings confirm that increasing the frequency of oscillation weakens the fluid velocity and the drag on both walls of the cylinders.

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Simultaneous Contour Method of a Trigonometric Integral by Prudnikov et al.

Mathematics ◽

10.3390/math9121453 ◽

2021 ◽

Vol 9 (12) ◽

pp. 1453

Author(s):

Robert Reynolds ◽

Allan Stauffer

Keyword(s):

Closed Form ◽

Contour Method ◽

Substantial Part ◽

Exponential Functions ◽

Closed Form Solutions ◽

Definite Integrals ◽

Transcendental Functions ◽

Trigonometric Integral

In this present work we derive, evaluate and produce a table of definite integrals involving logarithmic and exponential functions. Some of the closed form solutions derived are expressed in terms of elementary or transcendental functions. A substantial part of this work is new.

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Sum of alternating-like series as definite integrals

International Journal of Mathematical Education in Science and Technology ◽

10.1080/0020739x.2021.1949060 ◽

2021 ◽

pp. 1-11

Author(s):

Sergio A. Carrillo

Keyword(s):

Definite Integrals

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On some definite integrals and a new method of reducing a function of spherical co-ordinates to a series of spherical harmonics

Proceedings of the Royal Society of London ◽

10.1098/rspl.1902.0068 ◽

1903 ◽

Vol 71 (467-476) ◽

pp. 97-101 ◽

Cited By ~ 1

Keyword(s):

Spherical Harmonics ◽

Fundamental Theorem ◽

New Method ◽

Definite Integrals

The expansion of a function f(θ) of an angle θ varying between 0 and π in terms of a series proceeding by the sines of the multiples of θ depends on the fundamental theorem, ∫ π 0 sin pθ sin qθ dθ = 0, where p and q are integer numbers not equal to each other.

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