Problems solving in the integral calculus and the determination of the area under the curve

Juan E. N´apoles Vald´es; Maria Nubia Quevedo

doi:10.37084/REMATEC.1980-3141.2024.n52.e2025005.id732

Autores

Juan E. N´apoles Vald´es Northeast Normal University https://orcid.org/0000-0003-2470-1090
Maria Nubia Quevedo Military University Nueva Granada https://orcid.org/0000-0002-4137-7408

DOI:

10.37084/REMATEC.1980-3141.2024.n52.e2025005.id732

Palavras-chave:

Problems solving, integral calculus, area under the curve

Resumo

In Integral Calculus the classic problem is the determination of the area under the curve, when said region is not expressible in terms of elementary figures. This translates into a multiplicity of problems and exercises that are presented to students in a Calculus course. This article presents a useful problem for Mathematics Education, derived from a generalized integral operator, for this we define what we understand by an integrable function in this generalized sense, and the geometric interpretation of a generalized definite integral is presented. The interesting thing about this generalization is that said geometric interpretation is similar to the geometric interpretation of the classical Riemann integral, but not in the xy plane, but in the Ty plane, where T is the kernel of the generalized integral.

Downloads

Não há dados estatísticos.

Métricas

Carregando Métricas ...

Biografia do Autor

Juan E. N´apoles Vald´es, Northeast Normal University

Faculty of Exact and Natural Sciences and Surveying

Maria Nubia Quevedo, Military University Nueva Granada

Universidad Militar Nueva Granada Facultad de Ciencias B´asicas y Aplicadas

Referências

Cristian Alfaro-Carvajal, Jennifer Fonseca-Castro, Problem solving in the teaching of single variable differential and integral calculus: Perspective of mathematics teachers, UNICIENCIA Vol. 32, N° 2, pp. 42-56. Julio-Diciembre, 2018 DOI: http://dx.doi.org/10.15359/ru.32-2.1 DOI: https://doi.org/10.15359/ru.32-2.3

I. R. S. Alves, M. C. Mancebo, T. C. S. Boncompagno, W. D. O. J’unior, E. C. Romao, and R. V. Garcia, Problem-Based Learning: A Tool for the Teaching of Definite Integral and the Calculation of Areas, International Journal of Information and Education Technology, Vol. 9, No. 8, August 2019 doi: 10.18178/ijiet.2019.9.8.1272 DOI: https://doi.org/10.18178/ijiet.2019.9.8.1272

B. D’Amore, Did’actica de la matem’atica, Editorial Magisterio, Bogot’a, 2021 ISBN 9789582014056

A. Arcavi, The role of visual representations in the learning of mathematics, Educational Studies in Mathematics, vol. 52, núm. 3, 2003, 215-24 DOI: https://doi.org/10.1023/A:1024312321077

G. Arrigo, B. D’Amore, S. Sbaragli, INFINITOS INFINITOS. Historia, filosof’ia y did’actica del infinito matem’atico, Colecciones: Did’acticas, Editorial: Magisterio, Bogot’a 2011

Awaludin, Basuki Wibawa, Murni Winarsih, Integral Calculus Learning Using Problem Based Learning Model Assisted by Hypermedia-Based E-Book, JPI, Vol. 9 No. 2, June 2020, 224-235 DOI: 10.23887/jpi-undiksha.v9i2.23106 DOI: https://doi.org/10.23887/jpi-undiksha.v9i2.23106

B. Balakrishnan, Exploring the impact of design thinking tool among design undergraduates: a study on creative skills and motivation to think creatively. Int J Technol Des Educ 2021. https://doi.org/10.1007/s10798-021-09652-y DOI: https://doi.org/10.1007/s10798-021-09652-y

R. E. Castillo, J. E. N’apoles Vald’es and H. C. Chaparro, OMEGA DERIVATIVE, Gulf Journal of Mathematics, Vol 16, Issue 1 (2024) 55-67 https://doi.org/10.56947/gjom.v16i1.1430 DOI: https://doi.org/10.56947/gjom.v16i1.1430

I. Cınar, On Some Properties of Generalized Riesz Potentials, Intern. Math. Journal, Vol. 3,2003, no. 12, 1393-1397

C. Dolores F., M. Garc’ia P., J. E. N’apoles V., J. M. Sigarreta A., AN APPROACH TO THE HISTORY OF MATHEMATICS, Far East Journal of Mathematical Education Volume 16, Issue 3, Pages 331 - 346 (August 2016). DOI: https://doi.org/10.17654/ME016030331

G. Farid, Study of a generalized Riemann-Liouville fractional integral via convex functions, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. Volume 69, Number 1, 37-48 (2020) DOI: 10.31801/cfsuasmas.484437 DOI: https://doi.org/10.31801/cfsuasmas.484437

A. Fleitas, J. E. Nápoles, J. M. Rodríguez, J. M. Sigarreta, NOTE ON THE GENERALIZED CONFORMABLE DERIVATIVE, Revista de la UMA, Volume 62, no. 2 (2021), 443-457

https://doi.org/10.33044/revuma.1930 DOI: https://doi.org/10.33044/revuma.1930

R. Gorenflo, F. Mainardi, Fractional Calculus: Integral and Differential Equations of Fractional Order, Springer, Wien (1997), 223-276. DOI: https://doi.org/10.1007/978-3-7091-2664-6_5

P. M. Guzmán, G. Langton, L. M. Lugo, J. Medina and J. E. Nápoles Valdés. A new definition of a fractional derivative of local type, J. Math. Anal. {bf 9:2} (2018), 88-98.

L. L. Helms, Introduction To Potential Theory (New York: Wiley-Interscience, 1969).

F. Hitt, Una reflexión sobre la construcción de conceptos matemáticos en ambientes con tecnología, Boletín de la Asociación Matemática Venezolana, vol. X, num. 2, 2003, 213-223

E. Jablonka, A. Klisinska, A note on the institutionalization of mathematical knowledge or, ``What was and is the Fundamental Theorem of Calculus, really?”, in B. Sriraman (Ed.) Crossroads in the History of Mathematics and Mathematical Education, The Montana Mathematics Enthusiast Monographs in Mathematics Education, Monograph 12, 2012

U. N. Katugampola, New approach to a generalized fractional integral, Appl. Math. Comput. 218 (2011), 860-865. DOI: https://doi.org/10.1016/j.amc.2011.03.062

U. N. Katugampola, A new approach to generalized fractional derivatives, Bull. Math. Anal. App. 6 (2014), 1–15.

M. N. Kholid, A. Imawati, A. Swastika, S. Maharani, L. N. Pradana, How are Students’ Conceptual Understanding for Solving Mathematical Problem? J Phys Conf Ser 2021;1776:012018. https://doi.org/10.1088/1742-6596/1776/1/012018 DOI: https://doi.org/10.1088/1742-6596/1776/1/012018

M. Kline, Matemáticas. La pérdida de la certidumbre, Editorial Siglo XXI de España, 2006 ISBN 968-23-1939-0

Frank Krueger, Maria Vittoria Spampinato, Matteo Pardini, Sinisa Pajevic, Jacqueline N. Wood, George H. Weiss, Ste¡en Landgraf, Jordan Grafman, Integral calculus problem solving: an fMRI investigation, NeuroReport 19(11), 2008, 1095-1099 DOI: 10.1097/WNR.0b013e328303fd85 DOI: https://doi.org/10.1097/WNR.0b013e328303fd85

Y. C. Kwun, G. Farid, W. Nazeer, S. Ullah, S. M. Kang, Generalized Riemann-Liouville k-fractional integrals associated with Ostrowski type inequalities and error bounds of Hadamard inequalities, IEEE Access, 6 (2018), 64946-64953. DOI: https://doi.org/10.1109/ACCESS.2018.2878266

R. Lesh, Directions for future research and development in engineering education, in J. Zawojewski, H. Diefes-Dux and K. Bowman (Eds.) Models and Modeling in Engineering Education: Designing Experiences for All Students. Rotterdam: Sense Publications, 2008.

G. M. Mittag-Leffler, Sur la nouvelle fonction, C. R. Acad. Sci. Paris,137,554-558, (1903).

G. M. Mittag-Leffler, Sur la représentation analytique d’une branche uniforme d’une fonction monogéne, Acta Math., Paris,29,101-181, (1904). DOI: https://doi.org/10.1007/BF02403200

C. Martinez, M. Sanz, F. Periogo, Distributional Fractional Powers of Laplacian, Riesz Potential. Studia Mathematica 135 (3) 1999.

J. E. Nápoles Valdés, De las cavernas a los fractales. Conferencias de historia de la Matemática, Universidad Pedagógica de Holguín, Cuba, 1996, 283 p. Editado por la Editorial de la Universidad Tecnológica Nacional, 2008 ISBN 978-987-26665-9-0, ISBN 978-987-27056-0-2.

J. E. Nápoles Valdés, La resolución de problemas en la escuela. Consejos preliminares, Revista Función Continua 8(2000), 21-42.

J. E. Nápoles Valdés, La resolución de problemas en la escuela. Algunas reflexiones, Educacao Matemática em Revista-RS, 2(2000), 51-65.

J. E. Nápoles Valdés, Los problemas. El hilo de Ariadna en la Historia de la Matemática, Revista del Instituto de Matem’atica, 6(12), 2010, 45-70.

J. E. Nápoles Valdés, Some reflections on mathematics and mathematicians. Simple questions, complex answers, The Mathematics Enthusiast, Vol. 9, nos.1 & 2, 2012, 221-232. DOI: https://doi.org/10.54870/1551-3440.1242

J. E. Nápoles Valdés, SOME REFLECTIONS ON THE PROBLEMS AND THEIR ROLE IN THE DEVELOPMENT OF MATHEMATICS, Qualitative Research Journal. São Paulo (SP), v.8, n.18, p. 524-539, ed. especial. 2020 524 Special Edition: Philosophy of Mathematics DOI: https://doi.org/10.33361/RPQ.2020.v.8.n.18.343

J. E. Nápoles Valdés, A. González Thomas, F. Genes, F. Basabilbaso, J. M. Brundo, El enfoque histórico-problémico en la enseñanza de la matemática para ciencias técnicas: el caso de las ecuaciones diferenciales ordinarias, Acta Scientae, V.6, N.2 (2004), 41-59.

J. E. Nápoles Valdés, P. M. Guzmán, L. M. Lugo, A. Kashuri, The local generalized derivative and Mittag Leffler function, Sigma J Eng & Nat Sci 38 (2), 2020, 1007-1017

P. Odifreddi, La matemática del siglo XX. De los conjuntos a la complejidad, Buenos Aires, Katz 2006 ISBN 987-1283-17-2

N. C. Presmeg, Research on visualization in learning and teaching mathematics, en Sense Publishers (ed.), Handbook of Research on the Psychology of Mathematics Education: Past, Present and Future, 2006, 205-235 DOI: https://doi.org/10.1163/9789087901127_009

N. K. Rahmawati, S. B. Waluya, Rochmad, I. Hidayah, METACOGNITIVE SKILLS ANALYSIS OF STUDENTS IN INTEGRAL CALCULUS PROBLEM SOLVING, JURNAL PENDIDIKAN MATEMATIKA DAN IPA Vol. 12, No. 2 (2021) 170-179 DOI: https://doi.org/10.26418/jpmipa.v12i2.45052

C. Sánchez, C. Valdés, De los Bernoulli a los Bourbaki. Una historia del arte y la ciencia del cálculo Ed. Nivola. Madrid, 2004ISBN: 84-95599-70-8

E. A. Silver, Teaching and learning mathematical problem solving: Multiple research perspectives. Teach Learn Math Probl Solving Mult Res Perspect 2013:1–469. https://doi.org/10.4324/9780203063545 DOI: https://doi.org/10.4324/9780203063545

B. Sriraman, L. English (Eds.), Theories of mathematics education: seeking new frontiers. (Springer series: advances in mathematics education), Springer Verlag Berlin Heidelberg, 2010 ISBN: 978-3-642-00741-5 DOI: https://doi.org/10.1007/978-3-642-00742-2

bibitem{W} A. Wiman, Über den fundamental satz in der theorie der funktionen , Acta Math.,29, 191- 201, (1905). DOI: https://doi.org/10.1007/BF02403202

D. Zhao, M. Luo, General conformable fractional derivative and its physical interpretation, Calcolo, 54: 903-917, 2017. DOI 10.1007/s10092-017-0213-8. DOI: https://doi.org/10.1007/s10092-017-0213-8

W. Zimmerman, S. Cunningham (eds.) (1991), Visualization in Teaching and Learning Mathematics, Estados Unidos, The Mathematical Association of America (Notes 19).