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

Un appaltatore deve stimare la quantità di vernice necessaria per coprire una specifica parte di un muro con un bordo superiore curvo in cento case modello. Per farlo con precisione, è necessario calcolare la superficie del muro.Se l'arco curvo segue una funzione matematica, il problema si riduce a trovare l'area sotto una data curva.Per approssimare quest'area, la regione sotto la curva è divisa in n numeri di rettangoli di larghezza Δx. La somma delle aree di questi rettangoli fornisce una stima dell'area totale.L'altezza di ogni rettangolo può essere misurata all'estremità sinistra o a destra, il che può portare a una sovrastima o sottostima a seconda della forma della curva.Una stima più bilanciata utilizza il valore della funzione in qualsiasi punto all'interno di ogni sottointervallo, chiamato punto campione.Per ogni rettangolo, l'area è data dal valore della funzione al punto campione moltiplicato per la larghezza del sottointervallo. Sommando le aree di tutti i rettangoli si ottiene l'area approssimativa.Man mano che il numero di rettangoli aumenta e la loro larghezza diminuisce, la somma si avvicina all'integrale, che fornisce l'area esatta sotto la curva. Questo aiuta a stimare la quantità esatta di vernice necessaria.

problema dell’area

Determinare l&#x2019;area di una regione con bordi rettilinei &#xE8; immediato, poich&#xE9; le formule geometriche per rettangoli, triangoli e poligoni possono essere applicate direttamente. Tuttavia, i metodi geometrici tradizionali non sono sufficienti quando una regione presenta un contorno curvilineo, come nel caso dell&#x2019;area sottesa dal grafico di una funzione.<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>da<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>Il problema dell&#x2019;area consiste nel trovare un criterio sistematico per misurare tali regioni. Un approccio per risolvere questo problema consiste nel ricorrere a un&#x2019;approssimazione. Invece di tentare di calcolare esattamente l&#x2019;area fin dall&#x2019;inizio, la regione sotto la curva viene dapprima suddivisa in forme pi&#xF9; piccole e pi&#xF9; semplici. Un metodo comune consiste nell&#x2019;approssimare l&#x2019;area con rettangoli. Sommando le aree di questi rettangoli si ottiene una stima dell&#x2019;area totale. L&#x2019;altezza di ciascun rettangolo &#xE8; determinata valutando la funzione in punti specifici dell&#x2019;intervallo. Scelte diverse di tali punti possono condurre a sovrastime o a sottostime dell&#x2019;area effettiva.All&#x2019;aumentare del numero di rettangoli e al ridursi della loro larghezza, l&#x2019;approssimazione diventa pi&#xF9; accurata. Al limite, quando la larghezza di ciascun rettangolo tende a zero, la somma delle loro aree converge a un valore esatto, che rappresenta l&#x2019;area effettiva sottesa dalla curva. Questo procedimento fornisce una base rigorosa per definire aree nei casi in cui intervengano contorni curvilinei.Il metodo di approssimazione delle regioni curve mediante la scomposizione in forme geometriche pi&#xF9; semplici va oltre la matematica ed &#xE8; ampiamente applicato in fisica, in economia e in ingegneria. Consente calcoli precisi in situazioni che riguardano quantit&#xE0; accumulate, come il lavoro compiuto da una forza variabile o i ricavi totali nel tempo.

Problema dell’area

Determinare l&#x2019;area di una regione con bordi rettilinei &#xE8; immediato, poich&#xE9; le formule geometriche per rettangoli, triangoli e poligoni ...

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Calculus

Analisi matematica

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Integrals

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KEY TERMS AND CONCEPTS

TERMINI E CONCETTI CHIAVE

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...

distance problem

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definite integral

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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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...

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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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substitution rule applied to definite integrals

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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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