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

Um empreiteiro precisa estimar a quantidade de tinta necessária para cobrir uma parte específica de uma parede com borda curva no topo em cem casas modelo. Para fazer isso com precisão, a área de superfície da parede deve ser calculada.Se a aresta curva segue uma função matemática, o problema se resume a encontrar a área sob uma dada curva.Para aproximar essa área, a região abaixo da curva é dividida em n números de retângulos, de largura Δx. A soma das áreas desses retângulos fornece uma estimativa da área total.A altura de cada retângulo pode ser tomada na extremidade esquerda ou direita, o que pode levar a uma superestimação ou subestimação dependendo do formato da curva.Uma estimativa mais equilibrada usa o valor da função em qualquer ponto dentro de cada subintervalo, chamado de ponto amostral.Para cada retângulo, a área é dada pelo valor da função no ponto de amostragem multiplicado pela largura do subintervalo. Somando as áreas de todos os retângulos, obtém-se a área aproximada.À medida que o número de retângulos aumenta e sua largura diminui, a soma se aproxima da integral, que fornece a área exata sob a curva. Isso ajuda a estimar a quantidade exata de tinta necessária.

problema da área

Determinar a &#xE1;rea de uma regi&#xE3;o com bordas retas &#xE9; simples, pois as f&#xF3;rmulas geom&#xE9;tricas para ret&#xE2;ngulos, tri&#xE2;ngulos e pol&#xED;gonos podem ser aplicadas diretamente. No entanto, os m&#xE9;todos geom&#xE9;tricos tradicionais s&#xE3;o insuficientes quando uma regi&#xE3;o possui uma borda curva, como a &#xE1;rea sob uma fun&#xE7;&#xE3;o.<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>a partir 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>O problema da &#xE1;rea envolve encontrar uma maneira sistem&#xE1;tica de medir tais regi&#xF5;es. Uma abordagem para resolver esse problema &#xE9; por meio de aproxima&#xE7;&#xE3;o. Em vez de tentar calcular a &#xE1;rea exatamente desde o in&#xED;cio, a regi&#xE3;o sob a curva &#xE9; primeiro dividida em formas menores e mais simples. Um m&#xE9;todo comum envolve aproximar a &#xE1;rea usando ret&#xE2;ngulos. Somando as &#xE1;reas desses ret&#xE2;ngulos, obtemos uma estimativa da &#xE1;rea total. A altura de cada ret&#xE2;ngulo &#xE9; determinada avaliando a fun&#xE7;&#xE3;o em pontos espec&#xED;ficos ao longo do intervalo. Escolhas diferentes para esses pontos podem levar a superestimativas ou subestimativas da &#xE1;rea real.&#xC0; medida que o n&#xFA;mero de ret&#xE2;ngulos aumenta e suas larguras diminuem, a aproxima&#xE7;&#xE3;o torna-se mais precisa. No limite, &#xE0; medida que a largura de cada ret&#xE2;ngulo tende a zero, a soma de suas &#xE1;reas converge para um valor exato, que representa a verdadeira &#xE1;rea sob a curva. Esse processo fornece uma base rigorosa para a defini&#xE7;&#xE3;o de &#xE1;reas em casos que envolvem contornos curvos.O m&#xE9;todo de aproximar regi&#xF5;es curvas decompondo-as em formas geom&#xE9;tricas mais simples vai al&#xE9;m da matem&#xE1;tica e &#xE9; amplamente aplicado em f&#xED;sica, economia e engenharia. Ele permite c&#xE1;lculos precisos em cen&#xE1;rios que envolvem quantidades acumuladas, como o trabalho realizado por uma for&#xE7;a vari&#xE1;vel ou a receita total ao longo do tempo.

Problema da área

Determinar a &#xE1;rea de uma regi&#xE3;o com bordas retas &#xE9; simples, pois as f&#xF3;rmulas geom&#xE9;tricas para ret&#xE2;ngulos, tri&#xE2;ngulos e ...

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Calculus

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Integrals

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KEY TERMS AND CONCEPTS

TERMOS E CONCEITOS-CHAVE

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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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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