INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE
ISSN: 2692-5206, Impact Factor: 12,23
American Academic publishers, volume 05, issue 05,2025
Journal:
https://www.academicpublishers.org/journals/index.php/ijai
page 410
APPLICATIONS OF DEFINITE INTEGRALS IN MECHANICS
Yoʻldosheva Durdona Botirboy kizi
Urgench State Pedagogical Institute, Faculty of Exact and Applied Sciences,
1st-year student of the Mathematics program.
Jumanazarova Shodiya Otojon kizi
Urgench State Pedagogical Institute
Teacher of the department of physics and Mathematics
Abstract:
This article explores the application of definite integrals in solving mechanical
problems. In particular, the determination of the center of gravity, moment of inertia,
rectangle approximation formulas, and numerical integration methods such as the trapezoidal
and Simpson’s rules are discussed. As an essential tool in mathematical analysis, the definite
integral is used to carry out computations on real physical objects.
Keywords:
definite integral, center of gravity, moment of inertia, trapezoidal rule, Simpson's
rule, static moment, mechanics, integral calculation
Introduction
The theory of definite integrals, a core part of mathematical analysis, finds extensive
application in natural sciences, especially in physics and mechanics. Solving basic
mechanical problems such as finding the center of gravity, moment of inertia, and force
distribution requires the use of integrals. For bodies of complex shapes, analytical methods
may not be feasible; hence, numerical integration and approximate solutions become essential.
Literature Review
There are numerous reputable sources on the application of definite integrals. For example,
'Mathematical Analysis' by R. G. Bartle and D. R. Sherbert discusses the theoretical
foundations and physical applications of definite integrals. The Uzbek-language textbook
'Matematical Analysis' by A. M. Mamatov and T. T. Eshmamatov provides a comprehensive
explanation of center of gravity and moment of inertia in integral terms. Moreover, integral
formulas are extensively covered in engineering and physics literature.
Main Part (Outlined Topics)
1. Static Moment and Center of Gravity
The static moment reflects how mass or area is distributed with respect to a certain axis. The
center of gravity is the point at which the entire mass of a div appears to be concentrated.
a) Center of Gravity of a Plane Curve
INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE
ISSN: 2692-5206, Impact Factor: 12,23
American Academic publishers, volume 05, issue 05,2025
Journal:
https://www.academicpublishers.org/journals/index.php/ijai
page 411
Determined by the following formulas:
x̄ = (1/L) ∫C x ds, y = (1/L) ∫C y ds
where ds = √(1 + (dy/dx)^2) dx and L is the total arc length.
b) Center of Gravity of a Plane Shape
Defined over the area:
x̄ = (1/A)
∬
D x dA, y = (1/A)
∬
D y dA
where A is the total area.
2. Moment of Inertia
The moment of inertia about a given axis is:
Ix = ∫ y² dm, Iy = ∫ x² dm
For plane regions:
Ix = ∫[a to b] y²ρ(x) dx
where ρ(x) is the density function.
3. Rectangle Approximation Formulas
The integral can be approximated using the rectangle rule:
∫[a to b] f(x) dx ≈ Σ f(xi*) Δx
where xi* is a point in each subinterval and Δx = (b - a)/n.
4. Numerical Integration of Definite Integrals
When exact analytical computation is difficult or impossible, numerical methods are
employed. Trapezoidal and Simpson's rules are among the most effective.
5. Trapezoidal Rule
∫[a to b] f(x) dx ≈ (h/2) [f(a) + 2 Σ f(xi) + f(b)]
where h = (b-a)/n; this formula estimates the area using trapezoids.
6. Simpson’s (Parabolic) Rule
∫[a to b] f(x) dx ≈ (h/3) [f(a) + 4 Σ (odd indices) f(xi) + 2 Σ (even indices) f(xi) + f(b)]
This method approximates the function using parabolas and yields high accuracy.
Discussion
Using the above methods, it is possible to calculate the physical characteristics of complex-
shaped bodies such as their center of gravity and moment of inertia. The trapezoidal and
Simpson's rules are widely used both theoretically and in practical computer applications. In
fields like engineering and architecture, definite integrals play a central role in performing
design and structural calculations.
Results
- Definite integrals serve as the main mathematical tool in determining the center of gravity
and moment of inertia.
- Approximate methods like the trapezoidal and Simpson's rules are effective in calculating
integrals of complex functions.
INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE
ISSN: 2692-5206, Impact Factor: 12,23
American Academic publishers, volume 05, issue 05,2025
Journal:
https://www.academicpublishers.org/journals/index.php/ijai
page 412
- Integral-based analysis is critical in engineering design and mechanical planning.
Conclusion
This article analyzed key applications of definite integrals in mechanics. It was demonstrated
that concepts such as static moment, center of gravity, and moment of inertia can be precisely
calculated using integration. The advantages of numerical methods such as the trapezoidal
and Simpson’s rules were also discussed. It is evident that definite integrals are indispensable
in the accurate analysis of physical and mechanical systems.
References:
1. Mamatov A. M., Eshmamatov T. T. — Matematical Analysis, Tashkent, 2008.
2. Bartle R. G., Sherbert D. R. — Introduction to Real Analysis, Wiley, 2011.
3. Rozhdestvensky E. A. — Course of Higher Analysis, Moscow, 2006.
4. Stewart J. — Calculus: Early Transcendentals, Brooks/Cole, 2015.
5. Berezansky Yu. M. — Difference and Integral Equations, Kyiv, 1983.
