Name: definite integral rules for motion
Uploaded: 2026-01-12T12:24:01.000Z
Duration: 1 min 17 s
Channel: JoVE

L’intégrale définie calcule des grandeurs comme le déplacement parcouru.Lors de la comparaison de deux voitures se déplaçant avec des vitesses différentes, on utilise les propriétés de positivité et de comparaison de l’intégrale.Considérons la vitesse de la voiture v(t) sur l’intervalle de temps [a,b].Comme la voiture ne recule jamais, la vitesse v(t) est non négative sur [a,b].L’intégrale définie donne le déplacement parcouru. Puisque la fonction est non négative, la valeur de l’intégrale est également non négative, ce qui montre la propriété de positivité des intégrales.Imaginons maintenant deux voitures, la voiture 1 et la voiture 2, avec les fonctions de vitesse f(t) et g(t), se déplaçant sur le même intervalle.Si f(t) est plus grand ou égal à g(t) à chaque fois t, alors la voiture 1 doit parcourir une cylindrée supérieure ou égale à la voiture 2.Ainsi, l’intégrale définie de f(t) sur l’intervalle de temps [a,b] est supérieure ou égale à l’intégrale définie de g(t), ce qui montre la propriété de comparaison des intégrales.

propriétés des intégrales définies iii

L&#x2019;int&#xE9;grale d&#xE9;finie joue un r&#xF4;le fondamental dans la compr&#xE9;hension du mouvement, en particulier lorsqu&#x2019;il s&#x2019;agit de calculer la distance parcourue par un objet au cours du temps. Deux principes essentiels d&#xE9;coulent de cette application : la propri&#xE9;t&#xE9; de positivit&#xE9; et la propri&#xE9;t&#xE9; de comparaison des int&#xE9;grales d&#xE9;finies. Ces propri&#xE9;t&#xE9;s fournissent des interpr&#xE9;tations physiques intuitives fond&#xE9;es sur la vitesse et le d&#xE9;placement.
Propri&#xE9;t&#xE9; de positivit&#xE9; des int&#xE9;grales d&#xE9;finies
La propri&#xE9;t&#xE9; de positivit&#xE9; affirme que si la vitesse d&#x2019;un objet demeure non n&#xE9;gative &#x2014; c&#x2019;est-&#xE0;-dire si l&#x2019;objet ne se d&#xE9;place jamais vers l&#x2019;arri&#xE8;re &#x2014; alors le d&#xE9;placement total calcul&#xE9; sur un intervalle de temps donn&#xE9; doit &#xE9;galement &#xEA;tre non n&#xE9;gatif. En termes concrets, si une voiture se d&#xE9;place toujours vers l&#x2019;avant ou reste &#xE0; l&#x2019;arr&#xEA;t pendant un trajet, la distance qu&#x2019;elle parcourt ne peut pas &#xEA;tre n&#xE9;gative. Dans ce contexte, l&#x2019;int&#xE9;grale d&#xE9;finie additionne l&#x2019;ensemble des mouvements vers l&#x2019;avant, garantissant ainsi que le r&#xE9;sultat final corresponde &#xE0; une mesure r&#xE9;aliste et non n&#xE9;gative du d&#xE9;placement.
Propri&#xE9;t&#xE9; de comparaison des int&#xE9;grales d&#xE9;finies
La propri&#xE9;t&#xE9; de comparaison prolonge cette id&#xE9;e en permettant de comparer le mouvement de deux objets sur le m&#xEA;me intervalle de temps. Supposons que deux voitures circulent sur une route et que la voiture 1 roule toujours &#xE0; une vitesse &#xE9;gale ou sup&#xE9;rieure &#xE0; celle de la voiture 2. Dans ce cas, la voiture 1 parcourra au moins la m&#xEA;me distance que la voiture 2 sur la m&#xEA;me dur&#xE9;e. Cette propri&#xE9;t&#xE9; montre que, si un objet se d&#xE9;place syst&#xE9;matiquement plus rapidement qu&#x2019;un autre, son d&#xE9;placement cumul&#xE9; doit n&#xE9;cessairement &#xEA;tre sup&#xE9;rieur ou &#xE9;gal. L&#x2019;int&#xE9;grale d&#xE9;finie rend compte de cette comparaison en additionnant les vitesses au cours du temps, ce qui refl&#xE8;te l&#x2019;&#xE9;cart de distance totale parcourue.
Pris ensemble, ces propri&#xE9;t&#xE9;s permettent d&#x2019;&#xE9;tablir un lien entre le concept abstrait d&#x2019;int&#xE9;gration et le mouvement dans le monde r&#xE9;el, en soulignant comment les int&#xE9;grales constituent des outils essentiels pour quantifier l&#x2019;accumulation des variations dans les syst&#xE8;mes physiques.

Propriétés des intégrales définies III

L&#x2019;int&#xE9;grale d&#xE9;finie joue un r&#xF4;le fondamental dans la compr&#xE9;hension du mouvement, en particulier lorsqu&#x2019;il s&#x2019;agit ...

Education

Enseignement

JoVE Core

Calculus

Calcul

Unit

Unité

Integrals

Intégrales

KEY TERMS AND CONCEPTS

TERMES ET CONCEPTS CLÉS

Properties of Definite Integral III

definite integral rules for motion

The definite integral helps describe motion by showing how far an object travels over time. In this lesson, the key ideas are the Positivity Property and the Comparison Property of definite integrals. Both ideas connect integration to velocity and displacement in a way that is easy to interpret in real situations.

The Positivity Property says that if an object’s velocity is nonnegative, then its total displacement over the time interval is also nonnegative. Nonnegative velocity means the object does not move backward. For example, if a car keeps moving forward or stays still during a trip, the distance it covers cannot be negative. The definite integral adds up the forward motion and gives a nonnegative result.

The Comparison Property lets us compare the motion of two objects over the same time interval. If Car 1 always moves at a speed equal to or greater than Car 2, then Car 1 travels at least as far as Car 2. This property shows that a consistently faster object has greater or equal accumulated displacement. The definite integral captures that difference by summing each velocity over time.

Together, these properties make integrals more useful for understanding physical systems. They show how a math tool can measure accumulated change in motion. They also help students connect the abstract idea of integration with real-world travel.

Definite Integral Rules for Motion

area problem

Approximating Area Under a Curve

Approximating area under a curve is a key way to measure regions with curved boundaries. Straight-sided shapes are easy to handle with formulas for rectangles, triangles, and polygons. Curved regions need a different method because standard geometry does not work well on them. The area problem solves this by breaking the region into smaller, simpler shapes. A common approach uses rectangles to estimate the space under a function. Each rectangle gets its height from the function at a chosen...

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Estimating Distance from Changing Velocity

Estimating distance from changing velocity uses small time intervals and velocity values. When an object speeds up or slows down, its total displacement can be found by adding many short pieces of motion. This method gives an approximation first, and integral calculus gives the exact result later. A runner who accelerates during the first three seconds of a race shows how the idea works. Velocity can be measured every half-second and multiplied by the 0.5-second time step. Using left-endpoint...

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Riemann Sums and Area Estimates

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Gauss Sums for Riemann Rectangles

Riemann sums use rectangles to estimate the area under a curve in definite integration. The interval is split into subintervals, and the rectangle areas are added together. When those rectangle heights follow a regular pattern, sum formulas can make the calculation faster and more exact. This page focuses on sums that appear in arithmetic and polynomial sequences. The sum of consecutive integers, squares, and cubes is especially useful when the partition is uniform or when the data has a...

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Estimating Area with the Midpoint Rule

The Midpoint Rule estimates the area under a curve by using rectangles. It is a numerical method that helps when an exact area is hard to find or not practical to calculate. By using rectangle heights from midpoints, it gives a useful approximation over a chosen interval. To use the Midpoint Rule, the interval is split into equal subintervals. The midpoint of each subinterval is found by averaging its two endpoints. The function value at that midpoint gives the height of a rectangle over that...

properties of definite integral i

Definite Integrals for Car Motion

Definite integrals help describe a car’s motion over time by linking velocity to displacement. In a velocity-time graph, the definite integral shows how velocity builds up over a chosen time interval. The area under the curve gives the total displacement, so the graph becomes a clear picture of how far the car moves. One key idea is linearity of definite integrals. This property lets you add or subtract velocity functions while keeping their accumulated effect. If a car’s motion is made of...

properties of definite integral ii

Definite Integrals in Motion Analysis

Definite integrals can be used to find displacement from a velocity-time graph. A velocity function v(t) describes how motion changes over time. The definite integral of v(t) from a to b gives the net displacement between those times. It also represents the signed area under the velocity curve between the two points. Two useful properties make definite integrals easier to work with in motion problems. The additivity property says that if a continuous function is integrated over adjacent...

properties of definite integral iii

The definite integral helps describe motion by showing how far an object travels over time. In this lesson, the key ideas are the Positivity Property and the Comparison Property of definite integrals. Both ideas connect integration to velocity and displacement in a way that is easy to interpret in real situations. The Positivity Property says that if an object’s velocity is nonnegative, then its total displacement over the time interval is also nonnegative. Nonnegative velocity means the...

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Definite Integrals: A Step-by-Step Strategy

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Accumulation and Rates in the Fundamental Theorem

The Fundamental Theorem of Calculus connects accumulation and rate of change. In this lesson, the idea is shown with a water tank that is filled by a pump. The pump rate changes over time, so the total water collected must be found by adding up the changing inflow. In water management, this kind of calculation matters for pressure control, operation timing, and system safety. The flow rate is given by a time-dependent function. To find how much water has entered the tank from the start to a...

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The Fundamental Theorem of Calculus, Part 2, connects definite integrals with antiderivatives. It shows that the area under a continuous curve can be found without adding infinitely many tiny rectangles one by one. Instead, the integral can be evaluated directly from an antiderivative. If a function f(x) is continuous on a closed interval [a, b], then its definite integral over that interval can be computed using any antiderivative F(x). The result is written as F(b) - F(a). This works because...

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Finding Water Volume with Integration

Integration can be used to find the total volume of water in a tank when the inflow rate changes over time. In this example, the pump starts at 5 L/min. The inflow rate then increases by 2 L/min for each additional minute. That changing rate can be written as a linear function. To measure how much water has been added, the inflow rate function is integrated over time. The total water volume V(t) comes from this integral. When the power rule is used, the result includes both a quadratic term...

net change theorem

Calculus for Reservoir Overflow Prediction

The Net Change Theorem links a function’s rate of change to its total accumulated change over an interval. In calculus, this means the definite integral of a derivative on [a,b] gives the net change in the original function. The theorem is useful in physics, economics, and engineering because it turns a rate into a total amount. One clear application is a reservoir that receives water from two inflow sources at the same time. The reservoir starts with 260 million cubic meters of water. Each...

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Work Done on a Nonlinear Spring

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Using Substitution to Evaluate Definite Integrals

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key terms and concepts

Learn how the Positivity Property and Comparison Property of definite integrals describe motion, velocity, and displacement in real-world examples.

Definite Integral Rules for Motion | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Definite Integral Rules for Motion

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