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Integration 3
Published in C.W. Evans, Engineering Mathematics, 2019
It is possible to use these ideas to extend the definition further so that infinite integrals may be considered. An infinite integral occurs when either the upper limit of integration is ∞ or the lower limit of integration is −∞, or both. For example, if the integral of f(x) with respect to x exists throughout the interval [0, ∞) then ∫0∞f(x)dx=limx→∞F(x)−F(0)
Numerical Integration
Published in William Bober, ® Essentials, 2017
An integral is improper if the integrand approaches infinity at some point within the limits of integration, including the end points. In many cases, the integration will still result in a finite solution. An example of an improper integral follows: () I=∫01log(1+x)xdx
Maple
Published in Paul W. Ross, The Handbook of Software for Engineers and Scientists, 2018
Definite integration is also implemented, and there are three different numerical algorithms to choose from. The limits of integration may be finite or infinite (by using the Maple constant infinity). As shown in the examples below, you can specify ranges in Maple by using two dots such as 0..infinity or a..b. Sometimes Maple will express the results of integration in terms of special functions such as the gamma function, the error function, or elliptic functions.
Intra-mathematical connections made by high school students in performing Calculus tasks
Published in International Journal of Mathematical Education in Science and Technology, 2018
Javier García-García, Crisólogo Dolores-Flores
This connection indicates a relationship between a mathematical concept (integral) and its corresponding meaning in the graphic register (area), but in particular, under a curve in the Cartesian plane (graphic). Three students established this connection (12%); it is possible that they are thinking of the Riemann sums (see excerpt from E12 and Figure 2). Interviewer: Do you know what is the integral?E12: Yes, it's the area under a curve.Interviewer: Does it have another meaning?E12: Well, I think that is to calculate area by sum areas of rectangles inscribed under the curve.The limits of integration mean the interval where you should calculate the area under the curve
Study on fractional-order elastic-plastic constitutive model of river silt based on critical state theory
Published in Marine Georesources & Geotechnology, 2022
Yuke Wang, Junhao Li, Xiang Yu, Xiaoying Lin, Jinggan Shao, Jiancang Zhao
In the formula, represents the -order differentiation of which can be expressed as in this paper. 0 and x are the lower and upper limits of integration respectively. From formula (4), it can be seen that the equation considers the starting and ending states in the loading process, and on the right side of the equation is a gamma function, with the specific form as follows: