Name: estimating area with the midpoint rule
Uploaded: 2026-01-12T12:24:26.000Z
Duration: 1 min 20 s
Channel: JoVE

A regra do ponto médio ajuda a estimar a área sob uma curva dividindo o intervalo em subintervalos iguais.O ponto médio de cada subintervalo é calculado como a média de seus dois pontos finais. O valor da função nesses pontos médios indica a altura dos retângulos desenhados sobre eles. Essa abordagem frequentemente fornece uma estimativa mais precisa do que métodos que usam extremidades esquerda ou direita, pois reduz erros causados por subestimação ou superestimação da área.Considere uma região ao lado de um rio. É delimitado por duas linhas verticais nas laterais, uma linha horizontal abaixo e uma margem curva acima.Para estimar sua área, a região é dividida em vários retângulos de largura igual.A altura de cada retângulo é encontrada usando a função no ponto médio de sua base.A área de cada retângulo é encontrada multiplicando sua altura pela largura do subintervalo. Essas áreas são somadas para aproximar a área total sob a curva, fornecendo a fórmula da regra do ponto médio.À medida que o número de retângulos aumenta e eles se tornam mais estreitos, a estimativa melhora e se aproxima da área real.

regra do ponto médio

Aproximar &#xE1;reas sob fronteiras curvadas &#xE9; um problema comum em matem&#xE1;tica aplicada, especialmente quando um c&#xE1;lculo exato &#xE9; dif&#xED;cil ou impratic&#xE1;vel. Um m&#xE9;todo num&#xE9;rico eficaz para esse prop&#xF3;sito &#xE9; a Regra do Ponto M&#xE9;dio, que fornece uma estimativa da &#xE1;rea sob uma curva por meio de aproxima&#xE7;&#xF5;es retangulares em um intervalo especificado.
Descri&#xE7;&#xE3;o da Regra do Ponto M&#xE9;dio
A Regra do Ponto M&#xE9;dio come&#xE7;a dividindo o intervalo dado em v&#xE1;rios subintervalos de igual largura. Para cada subintervalo, o ponto m&#xE9;dio &#xE9; identificado pela m&#xE9;dia dos valores entre seus dois pontos extremos. O valor da fun&#xE7;&#xE3;o nesse ponto m&#xE9;dio &#xE9; ent&#xE3;o utilizado para determinar a altura de um ret&#xE2;ngulo desenhado sobre o subintervalo. Como o ponto m&#xE9;dio geralmente se encontra mais pr&#xF3;ximo do comportamento m&#xE9;dio da fun&#xE7;&#xE3;o nesse intervalo, esse m&#xE9;todo tende a reduzir o erro associado &#xE0; superestima&#xE7;&#xE3;o ou subestima&#xE7;&#xE3;o que pode ocorrer quando as extremidades &#xE0; esquerda ou &#xE0; direita s&#xE3;o usadas.
Interpreta&#xE7;&#xE3;o Geom&#xE9;trica
Um exemplo pr&#xE1;tico da Regra do Ponto M&#xE9;dio pode ser ilustrado estimando-se a &#xE1;rea de uma regi&#xE3;o adjacente a um rio. A regi&#xE3;o &#xE9; delimitada &#xE0; esquerda e &#xE0; direita por linhas verticais, abaixo por uma linha horizontal e acima por uma margem curva do rio. Como o limite superior &#xE9; irregular, sua &#xE1;rea n&#xE3;o pode ser facilmente calculada por f&#xF3;rmulas geom&#xE9;tricas simples.
Processo de Aproxima&#xE7;&#xE3;o da &#xC1;rea
Para aproximar a &#xE1;rea dessa regi&#xE3;o, ela &#xE9; dividida em v&#xE1;rios ret&#xE2;ngulos de largura igual. A altura de cada ret&#xE2;ngulo &#xE9; determinada avaliando-se a curva de contorno no ponto m&#xE9;dio da base do ret&#xE2;ngulo. A &#xE1;rea de cada ret&#xE2;ngulo &#xE9; calculada multiplicando-se sua altura por sua largura. A soma das &#xE1;reas de todos os ret&#xE2;ngulos fornece uma estimativa da &#xE1;rea total sob a curva, que &#xE9; a ess&#xEA;ncia da Regra do Ponto M&#xE9;dio.
Precis&#xE3;o do M&#xE9;todo
&#xC0; medida que o n&#xFA;mero de ret&#xE2;ngulos aumenta e suas larguras diminuem, a aproxima&#xE7;&#xE3;o retangular segue mais de perto o formato da curva. Consequentemente, a &#xE1;rea estimada melhora e se aproxima da &#xE1;rea real da regi&#xE3;o.

Regra do Ponto Médio

Aproximar &#xE1;reas sob fronteiras curvadas &#xE9; um problema comum em matem&#xE1;tica aplicada, especialmente quando um c&#xE1;lculo exato &#xE9; ...

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KEY TERMS AND CONCEPTS

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Midpoint Rule

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 subinterval.

This midpoint choice often matches the average behavior of the curve better than using the left or right endpoint. That can reduce the error that may happen with overestimation or underestimation. In this way, the method gives a closer area estimate than simple endpoint-based rectangles in many cases.

The idea can be shown with a region next to a river. The region is bounded on the left and right by vertical lines, below by a horizontal line, and above by a curved riverbank. Since the upper edge is irregular, its area is not easy to find with simple geometry.

The area is approximated by dividing the region into several rectangles of equal width. Each rectangle’s area is found by multiplying its width by its midpoint height. Adding the areas of all the rectangles gives the estimated total area under the curve.

The estimate improves as the number of rectangles increases and their widths get smaller. The rectangles then follow the curve more closely. As a result, the approximation approaches the true area of the region.

Estimating Area with the Midpoint Rule

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key terms and concepts

Learn how the Midpoint Rule estimates area under a curve with equal-width rectangles, midpoint heights, and better accuracy as intervals get smaller.

Estimating Area with the Midpoint Rule | 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

Estimating Area with the Midpoint Rule

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