Perbandingan Penyelesaian Integral Riemann, Lebesgue dan HK Berdasarkan Definisi

Miftahul Fikri, Samsurizal Samsurizal, Andi Makkulau


Sejak integral diperkenalkan oleh Newton dan Leibniz pada abad ke-17, alat ukur ini terus dilakukan kajian dan perumuman hingga saat ini. Terdapat tiga integral yang dikenal secara luas yaitu integral Riemann, integral Lebesgue dan integral HK. Pada masanya integral Riemann diterapkan untuk menyelesaikan berbagai persoalan tetapi kemudian diketahui memiliki keterbatasan. Integral Lebesgue muncul untuk mengatasi keterbatasan ini. Integral Lebesgue pun kemudian diketahui memiliki keterbatasannya tersendiri sehingga muncul integral Henstock-Kurzweil (integral HK) untuk mengatasi keterbatasan integral Lebesgue. Dalam banyak literatur seringkali untuk menyelesaikan soal integral tidak menggunakan definisi/ kerangka acuan yang sama sehingga sangat sulit memahami substansi maupun keterkaitan ketiga integral ini. Oleh karena itu, pada penelitian ini dilakukan perbandingan penyelesaian persoalan integral menggunakan definisi. Adapun dari empat persoalan integral yang dibahas, integral Riemann dapat menyelesaikan satu persoalan, integral Lebesgue dapat menyelesaikan dua persoalan dan integral HK dapat menyelesaikan seluruh persoalan yang dibahas.


integral Riemann; integral Lebesgue; integral Henstock-Kurzweil

Full Text:



E. Cassirer, “Newton and Leibniz,” Philos. Rev., vol. 52, no. 4, pp. 366–391, 1943.

I. N. Pesin, Classical And Modern Integration Theories. ACADEMIC PRESS, 1970.

P. Y. Lee and R. Vyborny, The integral an easy approach after Kurzweil and Henstock. Cambridge University Press, 2000.

N. Anjum and J. H. He, “Laplace transform: Making the variational iteration method easier,” Appl. Math. Lett., vol. 92, pp. 134–138, 2019.

P. Borghesani, P. Pennacchi, S. Chatterton, and R. Ricci, “The velocity synchronous discrete Fourier transform for order tracking in the field of rotating machinery,” Mech. Syst. Signal Process., vol. 44, no. 1–2, pp. 118–133, 2014.

X. Zhang, L. Liu, Y. Wu, and Y. Zou, “Existence and uniqueness of solutions for systems of fractional differential equations with Riemann–Stieltjes integral boundary condition,” Adv. Differ. Equations, vol. 2018, no. 1, 2018.

F. Santo Pedro, E. Esmi, and L. C. de Barros, “Calculus for linearly correlated fuzzy function using Fréchet derivative and Riemann integral,” Inf. Sci. (Ny)., vol. 512, no. 88882, pp. 219–237, 2020.

L. T. Yeong, Henstock-Kurzweil Integration On Euclidean Spaces. World Scientific, 2011.

D. Zeng, R. Zhang, S. Zhong, G. Yang, Y. Yu, and K. Shi, “Novel Lebesgue-integral-based approach to improved results for neural networks with additive time-varying delay components,” J. Franklin Inst., vol. 354, no. 16, pp. 7543–7565, 2017.

E. Dubon and A. San Antolín, “The Lebesgue differentiation theorem revisited,” Expo. Math., vol. 37, no. 3, pp. 322–332, 2019.

S. Gudder, “Quantum measure and integration theory,” J. Math. Phys., vol. 123509, 2009.

F. Móricz, “The Lebesgue summability of trigonometric integrals,” J. Math. Anal. Appl., vol. 390, no. 1, pp. 188–196, 2012.

R. A. Gordon, The integrals of Lebesgue, Denjoy, Perron, and Henstock. 1994.

R. Henstock, “A Riemann-Type Integral of Lebesgue Power,” Can. J. Math., vol. 20, no. x, pp. 79–87, 1968.

G. Ye and W. Liu, “The distributional Henstock–Kurzweil integral and applications,” Monatshefte fur Math., vol. 181, no. 4, pp. 975–989, 2016.

J. Malý and W. F. Pfeffer, “Henstock-Kurzweil integral on BV sets,” Math. Bohem., vol. 141, no. 2, pp. 217–237, 2016.

W. Liu, G. Ye, and D. Zhao, “The distributional Henstock-Kurzweil integral and applications II,” J. Nonlinear Sci. Appl., vol. 10, no. 1, pp. 290–298, 2017.

A. Pruthi, “Riemann integral vs. Lebesgue integral: A perspective view,” Adv. Math. Sci. J., vol. 9, no. 7, pp. 4505–4512, 2020.

W. Wojas and J. Krupa, “Familiarizing Students with Definition of Lebesgue Integral: Examples of Calculation Directly from Its Definition Using Mathematica,” Math. Comput. Sci., vol. 11, no. 3–4, pp. 363–381, 2017.

T. Kiria and G. Pantsulaia, “Calculation of Lebesgue integrals by using uniformly distributed sequences,” Trans. A. Razmadze Math. Inst., vol. 170, no. 3, pp. 402–409, 2016.

R. G. Bartle and Donald R. Sherbert, “Introduction to Real Analysis.” John Wiley & Sons, p. 1015024, 2011.

D. S. Kurtz and C. W. Swartz, Theories of Integration, vol. 13. World Scientific, 2012.

R. P. Agarwal, C. Flaut, and D. O’Regan, An Introduction to Real Analysis. 2018.

H. Wilcox and D. L. Myres, An Introduction to Lebesgue Integration and Fourier Series. ROBERT E. KRIEGER PUBLISHING COMPANY, INC., 1978.

H. L. Royden. and P. M. Fitzpatrick, Real Analysis. Prentice Hall, 2010.

R. G. Bartle, A Modern Theory of Integration. American Mathematical Society, 2001.

C. Swartz, Gauge Integral. World Scientific, 2001.



  • There are currently no refbacks.

Jumlah Kunjungan:

Creative Commons License
Limits: Journal Mathematics and its Aplications by Pusat Publikasi Ilmiah LPPM Institut Teknologi Sepuluh Nopember is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
Based on a work at