Name: riemann sum area estimates
Uploaded: 2026-01-12T12:24:45.000Z
Duration: 1 min 24 s
Channel: JoVE

A integração aproximada é usada quando o valor exato de uma integral definida não pode ser calculado.Isso normalmente ocorre em dois casos principais: quando a antiderivada da função é desconhecida ou não existe em forma fechada, e quando a função é composta por dados empíricos, como um conjunto de pontos discretos de um experimento, em vez de uma fórmula contínua.Nessas situações, integrais definidas são estimadas usando somas de Riemann, que dividem o intervalo em n subintervalos iguais de largura Δx.Para cada subintervalo, um retângulo é construído, com sua altura definida pelo valor da função em um ponto específico dentro desse subintervalo.Na aproximação do extremo esquerdo Ln, a altura de cada retângulo é definida pelo valor da função na extremidade esquerda.Se a função estiver aumentando, esse método subestima a área; Se a função está diminuindo, ela superestima.A aproximação do extremo direito Rn usa a extremidade direita de cada subintervalo. Esse método superestima a área se a função estiver aumentando e vice-versa.Esses métodos usam geometria básica — adicionando áreas retangulares — para estimar integrais de funções complexas ou desconhecidas.

integração aproximada

Em muitos contextos pr&#xE1;ticos e te&#xF3;ricos, o valor exato de uma integral definida pode ser inacess&#xED;vel. Essa limita&#xE7;&#xE3;o geralmente ocorre quando a antiderivada de uma fun&#xE7;&#xE3;o &#xE9; desconhecida ou n&#xE3;o pode ser expressa em forma fechada. Alternativamente, isso pode ocorrer quando uma fun&#xE7;&#xE3;o &#xE9; definida n&#xE3;o por uma f&#xF3;rmula, mas por um conjunto finito de pontos de dados emp&#xED;ricos, como os coletados em experimentos. Nesses casos, as t&#xE9;cnicas de integra&#xE7;&#xE3;o aproximada oferecem uma solu&#xE7;&#xE3;o valiosa.
Uma das abordagens mais utilizadas baseia-se nas somas de Riemann, que estimam a &#xE1;rea sob a curva ao subdividir o intervalo de integra&#xE7;&#xE3;o em v&#xE1;rias partes iguais. Cada subintervalo &#xE9; associado a um ret&#xE2;ngulo cuja altura &#xE9; determinada pelo valor da fun&#xE7;&#xE3;o em um ponto escolhido dentro desse subintervalo. Essa interpreta&#xE7;&#xE3;o geom&#xE9;trica simples permite aproxima&#xE7;&#xF5;es num&#xE9;ricas quando m&#xE9;todos anal&#xED;ticos n&#xE3;o s&#xE3;o vi&#xE1;veis.
A escolha do ponto dentro de cada subintervalo leva a diferentes m&#xE9;todos. Na aproxima&#xE7;&#xE3;o pelo ponto da extremidade &#xE0; esquerda, a altura de cada ret&#xE2;ngulo &#xE9; baseada no valor da fun&#xE7;&#xE3;o na extremidade esquerda do subintervalo. Quando a fun&#xE7;&#xE3;o &#xE9; crescente, esse m&#xE9;todo tende a subestimar a &#xE1;rea total; quando a fun&#xE7;&#xE3;o &#xE9; decrescente, ele tende a superestim&#xE1;-la. Por outro lado, a aproxima&#xE7;&#xE3;o pelo ponto extremo &#xE0; direita utiliza o valor da fun&#xE7;&#xE3;o na extremidade direita de cada subintervalo, resultando em uma superestima&#xE7;&#xE3;o para fun&#xE7;&#xF5;es crescentes e uma subestima&#xE7;&#xE3;o para fun&#xE7;&#xF5;es decrescentes.
Essas t&#xE9;cnicas, embora elementares em sua formula&#xE7;&#xE3;o, servem como ferramentas fundamentais na an&#xE1;lise num&#xE9;rica. Elas s&#xE3;o especialmente valiosas em aplica&#xE7;&#xF5;es cient&#xED;ficas e de engenharia, onde integrais precisam ser avaliadas a partir de medi&#xE7;&#xF5;es discretas ou em fun&#xE7;&#xF5;es complexas demais para integra&#xE7;&#xE3;o simb&#xF3;lica.

Integração aproximada

Em muitos contextos pr&#xE1;ticos e te&#xF3;ricos, o valor exato de uma integral definida pode ser inacess&#xED;vel. Essa limita&#xE7;&#xE3;o geralmente ...

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riemann sum area estimates

Riemann sums estimate the area under a curve when an exact definite integral is hard to find. This often happens when a function has no known antiderivative or cannot be written in a closed form. It also happens when the function comes from measured data instead of a formula.

The method works by splitting the interval of integration into equal parts. Each subinterval gets a rectangle, and the rectangle height comes from the function value at one chosen point in that part. Adding the rectangle areas gives an approximate integration result.

A left endpoint approximation uses the function value at the left end of each subinterval. For an increasing function, this method usually underestimates the total area. For a decreasing function, it usually overestimates it.

A right endpoint approximation uses the function value at the right end of each subinterval. For an increasing function, it usually overestimates the area. For a decreasing function, it usually underestimates it.

These rectangle-based estimates are simple, but they are useful in numerical analysis. They are especially helpful in science and engineering when an integral must be found from discrete measurements or when symbolic integration is not practical.

Riemann Sum Area Estimates

integration by parts: indefinite integrals

integration-by-parts-indefinite-integrals

Integration by Parts in AC Circuit Analysis

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integration by parts: definite integrals

Using Parts and Substitution for Area

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approximate integration

Riemann sums estimate the area under a curve when an exact definite integral is hard to find. This often happens when a function has no known antiderivative or cannot be written in a closed form. It also happens when the function comes from measured data instead of a formula. The method works by splitting the interval of integration into equal parts. Each subinterval gets a rectangle, and the rectangle height comes from the function value at one chosen point in that part. Adding the rectangle...

trapezoidal rule

Estimating Distance from Velocity Data

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simpson's rule i

Estimating Area with Simpson’s Rule

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improper integrals: discontinuous integrands

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Improper Integrals at an Endpoint

Improper integrals can appear when a definite integral has a discontinuity at one endpoint. The integrand may be undefined or become infinite there. In that case, the area under the curve is unbounded near the boundary. This behavior is often linked to a vertical asymptote at the edge of the interval. To evaluate this kind of integral, the endpoint is replaced with a variable. The integral is then written over an interval where the function is defined. After that, a limit is taken as the...

key terms and concepts

Learn how Riemann sums estimate area under a curve using left and right endpoint rectangles when exact integration is not available.

Riemann Sum Area Estimates | 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.

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Riemann Sum Area Estimates

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