Name: approximating area under a curve
Uploaded: 2026-01-12T12:24:24.000Z
Duration: 1 min 26 s
Channel: JoVE

Un entrepreneur doit estimer la quantité de peinture nécessaire pour recouvrir une partie spécifique d’un mur avec un bord supérieur courbé dans cent maisons miniatures. Pour faire cela avec précision, il faut calculer la surface du mur.Si l’arête courbe suit une fonction mathématique, le problème se réduit à trouver l’aire sous une courbe donnée.Pour approximer cette zone, la région sous la courbe est divisée en n nombre de rectangles de largeur Δx. La somme des aires de ces rectangles fournit une estimation de la surface totale.La hauteur de chaque rectangle peut être prise à l’extrémité gauche ou droite, ce qui peut entraîner une surestimation ou une sous-estimation selon la forme de la courbe.Une estimation plus équilibrée utilise la valeur de la fonction à n’importe quel point de chaque sous-intervalle, appelée point d’échantillonnage.Pour chaque rectangle, l’aire est donnée par la valeur de la fonction au point d’échantillon multiplicée par la largeur du sous-intervalle. En additionnant les aires de tous les rectangles, on obtient l’aire approximative.À mesure que le nombre de rectangles augmente et que leur largeur diminue, la somme tend vers l’intégrale, qui fournit l’aire exacte sous la courbe. Cela permet d’estimer la quantité exacte de peinture nécessaire.

problème d’aire

D&#xE9;terminer l&#x2019;aire d&#x2019;une r&#xE9;gion aux contours rectilignes est simple, car les formules g&#xE9;om&#xE9;triques applicables aux rectangles, aux triangles et aux polygones peuvent &#xEA;tre utilis&#xE9;es directement. En revanche, les m&#xE9;thodes g&#xE9;om&#xE9;triques traditionnelles se r&#xE9;v&#xE8;lent insuffisantes lorsqu&#x2019;une r&#xE9;gion pr&#xE9;sente une limite courbe, comme c&#x2019;est le cas de l&#x2019;aire situ&#xE9;e sous le graphe d&#x2019;une fonction.<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:25%;"><img style="aspect-ratio:300/24;" src="https://cloudfront.jove.com/files/ftp_upload/19973/19973_PT_Equation_1_editor_19973.svg" alt="19973_PT_Equation_1.svg" /></figure>de<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:25%;"><img style="aspect-ratio:300/24;" src="https://cloudfront.jove.com/files/ftp_upload/19973/19973_PT_Equation_2_editor_19973.svg" alt="19973_PT_Equation_2.svg" /></figure>Le probl&#xE8;me d&#x2019;aire consiste &#xE0; d&#xE9;terminer une m&#xE9;thode syst&#xE9;matique permettant de mesurer de telles r&#xE9;gions. Une approche pour r&#xE9;soudre ce probl&#xE8;me consiste &#xE0; recourir &#xE0; l&#x2019;approximation. Plut&#xF4;t que de chercher &#xE0; calculer l&#x2019;aire exactement d&#xE8;s le d&#xE9;part, la r&#xE9;gion situ&#xE9;e sous la courbe est d&#x2019;abord d&#xE9;coup&#xE9;e en figures plus petites et plus simples. Une m&#xE9;thode couramment utilis&#xE9;e consiste &#xE0; approcher l&#x2019;aire &#xE0; l&#x2019;aide de rectangles. En additionnant les aires de ces rectangles, vous obtenez une approximation de l&#x2019;aire totale. La hauteur de chaque rectangle est d&#xE9;termin&#xE9;e en &#xE9;valuant la fonction en des points pr&#xE9;cis de l&#x2019;intervalle. Selon le choix de ces points, l&#x2019;approximation peut conduire &#xE0; une surestimation ou &#xE0; une sous-estimation de l&#x2019;aire r&#xE9;elle.&#xC0; mesure que le nombre de rectangles augmente et que leur largeur diminue, l&#x2019;approximation devient de plus en plus pr&#xE9;cise. &#xC0; la limite, lorsque la largeur de chaque rectangle tend vers z&#xE9;ro, la somme de leurs aires converge vers une valeur exacte, laquelle repr&#xE9;sente l&#x2019;aire r&#xE9;elle sous la courbe. Ce processus fournit une base rigoureuse pour d&#xE9;finir les aires dans les situations o&#xF9; interviennent des bords courbes.La m&#xE9;thode consistant &#xE0; approximer des r&#xE9;gions courbes en les d&#xE9;composant en formes g&#xE9;om&#xE9;triques plus simples s&#x2019;&#xE9;tend au-del&#xE0; du domaine des math&#xE9;matiques et est largement utilis&#xE9;e en physique, en &#xE9;conomie et en ing&#xE9;nierie. Elle permet d&#x2019;effectuer des calculs pr&#xE9;cis dans des situations impliquant des quantit&#xE9;s cumul&#xE9;es, telles que le travail effectu&#xE9; par une force variable ou le revenu total au cours du temps.

Problème d’aire

D&#xE9;terminer l&#x2019;aire d&#x2019;une r&#xE9;gion aux contours rectilignes est simple, car les formules g&#xE9;om&#xE9;triques applicables aux ...

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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 point on the interval, and the areas of all the rectangles are then added together.

The choice of point can change the estimate. Some choices lead to an overestimate, while others lead to an underestimate of the true area. As more rectangles are used and each one becomes narrower, the estimate becomes more accurate.

In the limit, when the rectangle width approaches the smallest possible value, the sum of the rectangle areas reaches an exact value. That value is the true area under the curve. This method gives a rigorous way to measure curved regions and is also useful in physics, economics, and engineering, such as finding work done by a varying force or total revenue over time.

Approximating Area Under a Curve

area problem

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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Riemann Sums and Area Estimates

Riemann sums and area estimates are a key part of calculus for finding the area under a curve. When a real-valued function is defined on a closed interval, the area under its graph can be approximated if an exact calculation is not easy to do. The interval is split into equal subintervals. Each subinterval becomes a rectangle. The rectangle’s width is the length of the subinterval, and its height comes from the function value at one chosen point inside that part of the interval. Multiplying...

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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...

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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...

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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...

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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 are solved best with a clear, step-by-step strategy. The first step is to identify what the problem is asking for. The quantity may be accumulation, area, force, or probability. It is also important to note the integration limits and any constraints that affect the setup. Before evaluating the integral, look for properties that can simplify the work. If the expression has multiple parts, break it into smaller integrals. Known values from earlier work may help too. Symmetry...

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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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Definite Integrals with Antiderivatives

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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Integration in Breathing Cycle Modeling

Integration can be used to model the volume of air inhaled during breathing. The respiratory cycle is the repeating intake and expulsion of air, and it usually lasts about five seconds. By treating airflow as a function of time, students can see how the amount of air in the lungs changes during a breath. Airflow during respiration is not constant. It changes in a smooth, wave-like pattern, so a sinusoidal function is a good model. In a normal breathing cycle, the airflow rate reaches a maximum...

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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...

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Work Done on a Nonlinear Spring

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key terms and concepts

Learn how rectangle approximations help measure area under a curve, improve estimates, and lead to the exact value used in calculus and real-world applications.

Approximating Area Under a Curve | 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

Approximating Area Under a Curve

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