The Evaluation of ∫abdxx2 and ∫abdxx

1971 ◽  
Vol 64 (7) ◽  
pp. 605-606
Author(s):  
Norman Schaumberger
Keyword(s):  
Positive Integer ◽  
Fundamental Theorem ◽  
Definite Integral ◽  
Fundamental Theorem Of Calculus ◽  
Definition Of

Prior to the introduction of the fundamental theorem of calculus, it is a Common practice to use the definition of the definite integral to evaluate integrals of the form . These calculations are usually limited to cases in which k is a positive integer but are rarely extended to values larger than 3.

Technology Tips: Using Maple to Enhance Students' Understanding of Numerical Integration

2009 ◽  
Vol 103 (1) ◽  
pp. 76-80
Author(s):  
Nicole Scherger
Keyword(s):  
Numerical Integration ◽  
Fundamental Theorem ◽  
Definite Integral ◽  
Numerical Approximations ◽  
Fundamental Theorem Of Calculus ◽  
Calculus Students

Typically, calculus students are introduced to the simplest numerical approximations of the definite integral through the process of finding the areas of rectangles. Students are initially shown how to use the endpoints of each subinterval to find lower and upper sums, a process that gives them a bound on the actual area. They are then shown, sometimes through a series of labor-intensive computations or through visualization with graphs, that as the number of rectangles, or partitions, increases, the approximations become more and more accurate. Somewhere in this process students are probably also shown how to use midpoints to obtain slightly more accurate numerical approximations. At this point, most calculus courses lead students toward the fundamental theorem of calculus, at which time they learn that they can evaluate a definite integral by finding the antiderivative and evaluating between the limits of integration.

scholarly journals Fractional Line Integral

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1150
Author(s):  
Gabriel Bengochea ◽  
Manuel Ortigueira
Keyword(s):  
Fundamental Theorem ◽  
Piecewise Linear ◽  
Directional Derivatives ◽  
Definite Integral ◽  
Integral Calculus ◽  
Regular Curve ◽  
Line Integral ◽  
Linear Path ◽  
Definition Of

This paper proposed a definition of the fractional line integral, generalising the concept of the fractional definite integral. The proposal replicated the properties of the classic definite integral, namely the fundamental theorem of integral calculus. It was based on the concept of the fractional anti-derivative used to generalise the Barrow formula. To define the fractional line integral, the Grünwald–Letnikov and Liouville directional derivatives were introduced and their properties described. The integral was defined for a piecewise linear path first and, from it, for any regular curve.

scholarly journals Fractional Line Integral

2021 ◽  
Author(s):  
Gabriel Bengochea ◽  
Manuel Ortigueira
Keyword(s):  
Fundamental Theorem ◽  
Directional Derivatives ◽  
Definite Integral ◽  
Integral Calculus ◽  
Regular Curve ◽  
Line Integral ◽  
Definition Of

This paper proposes a definition of fractional line integral, generalising the concept of fractional definite integral. The proposal replicates the properties of the classic definite integral, namely the fundamental theorem of integral calculus. It is based on the concept of fractional anti-derivative used to generalise the Barrow formula. To define the fractional line integrals the Gr\"unwald-Letnikov and Liouville directional derivatives are introduced and their properties described. The integral is defined first for broken line paths and afterwards to any regular curve

Decomposing Complete 3-Uniform Hypergraph into 5-Cycles

2014 ◽  
Vol 672-674 ◽  
pp. 1935-1939
Author(s):  
Guan Ru Li ◽  
Yi Ming Lei ◽  
Jirimutu
Keyword(s):  
Positive Integer ◽  
Hamiltonian Cycles ◽  
Uniform Hypergraph ◽  
Design Algorithm ◽  
Uniform Hypergraphs ◽  
Edge Partition ◽  
Definition Of

About the Katona-Kierstead definition of a Hamiltonian cycles in a uniform hypergraph, a decomposition of complete k-uniform hypergraph Kn(k) into Hamiltonian cycles studied by Bailey-Stevens and Meszka-Rosa. For n≡2,4,5 (mod 6), we design algorithm for decomposing the complete 3-uniform hypergraphs into Hamiltonian cycles by using the method of edge-partition. A decomposition of Kn(3) into 5-cycles has been presented for all admissible n≤17, and for all n=4m +1, m is a positive integer. In general, the existence of a decomposition into 5-cycles remains open. In this paper, we use the method of edge-partition and cycle sequence proposed by Jirimutu and Wang. We find a decomposition of K20(3) into 5-cycles.

scholarly journals Integral transforms of the κ-Hilfer fractional derivative

2021 ◽  
Author(s):  
Felix Costa ◽  
Junior Cesar Alves Soares ◽  
Stefânia Jarosz
Keyword(s):  
Fractional Derivative ◽  
Gamma Function ◽  
Fundamental Theorem ◽  
Beta Function ◽  
Integral Transforms ◽  
Fractional Operators ◽  
Riesz Fractional Derivative ◽  
Hilfer Fractional Derivative ◽  
Fundamental Theorem Of Calculus

In this paper, some important properties concerning the κ-Hilfer fractional derivative are discussed. Integral transforms for these operators are derived as particular cases of the Jafari transform. These integral transforms are used to derive a fractional version of the fundamental theorem of calculus. Keywords: Integral transforms, Jafari transform, κ-gamma function, κ-beta function, κ-Hilfer fractional derivative, κ-Riesz fractional derivative, κ-fractional operators.

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