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

Un contratista debe estimar la cantidad de pintura necesaria para cubrir una parte específica de una pared con un borde superior curvo en cien casas modelo. Para hacerlo con precisión, es necesario calcular la superficie de la pared.Si la arista curva sigue una función matemática, el problema se reduce a encontrar el área bajo una curva dada.Para aproximar esta área, la región bajo la curva se divide en n números de rectángulos de ancho Δx. La suma de las áreas de estos rectángulos proporciona una estimación del área total.La altura de cada rectángulo puede tomarse en el extremo izquierdo o derecho, lo que puede llevar a una sobreestimación o una subestimación dependiendo de la forma de la curva.Una estimación más equilibrada utiliza el valor de la función en cualquier punto dentro de cada subintervalo, llamado punto muestral.Para cada rectángulo, el área se da por el valor de la función en el punto de muestra multiplicado por la anchura del subintervalo. Sumando las áreas de todos los rectángulos, se obtiene el área aproximada.A medida que aumenta el número de rectángulos y disminuye su anchura, la suma se acerca a la integral, que proporciona el área exacta bajo la curva. Esto ayuda a estimar la cantidad exacta de pintura necesaria.

problema del área

Determinar el &#xE1;rea de una regi&#xF3;n con lados rectos es sencillo, ya que las f&#xF3;rmulas geom&#xE9;tricas para rect&#xE1;ngulos, tri&#xE1;ngulos y pol&#xED;gonos se pueden aplicar directamente. Sin embargo, los m&#xE9;todos geom&#xE9;tricos tradicionales son insuficientes cuando una regi&#xF3;n tiene un l&#xED;mite curvo, como el &#xE1;rea bajo una funci&#xF3;n.<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>El problema del &#xE1;rea implica encontrar una forma sistem&#xE1;tica de medir dichas regiones. Un m&#xE9;todo para resolverlo es mediante la aproximaci&#xF3;n. En lugar de intentar calcular el &#xE1;rea con exactitud desde el principio, la regi&#xF3;n que se encuentra bajo la curva se divide primero en formas m&#xE1;s peque&#xF1;as y sencillas. Un m&#xE9;todo com&#xFA;n consiste en aproximar el &#xE1;rea utilizando rect&#xE1;ngulos. Al sumar las &#xE1;reas de estos rect&#xE1;ngulos, se obtiene una estimaci&#xF3;n del &#xE1;rea total. La altura de cada rect&#xE1;ngulo se determina evaluando la funci&#xF3;n en puntos espec&#xED;ficos a lo largo del intervalo. Diferentes opciones para estos puntos pueden llevar a sobreestimaciones o subestimaciones del &#xE1;rea real.A medida que aumenta el n&#xFA;mero de rect&#xE1;ngulos y se reduce su anchura, la aproximaci&#xF3;n se vuelve m&#xE1;s precisa. En el l&#xED;mite, a medida que la anchura de cada rect&#xE1;ngulo se aproxima a cero, la suma de sus &#xE1;reas converge a un valor l&#xED;mite, que representa el &#xE1;rea verdadera bajo la curva. Este proceso proporciona una base rigurosa para definir &#xE1;reas en casos donde existen l&#xED;mites curvos.El m&#xE9;todo de aproximaci&#xF3;n de regiones curvas mediante su descomposici&#xF3;n en formas geom&#xE9;tricas m&#xE1;s simples trasciende el &#xE1;mbito de las matem&#xE1;ticas y se aplica ampliamente en f&#xED;sica, econom&#xED;a e ingenier&#xED;a. Permite realizar c&#xE1;lculos precisos en escenarios que involucran cantidades acumuladas, como el trabajo realizado por una fuerza variable o los ingresos totales a lo largo del tiempo.

Problema del área

Determinar el &#xE1;rea de una regi&#xF3;n con lados rectos es sencillo, ya que las f&#xF3;rmulas geom&#xE9;tricas para rect&#xE1;ngulos, tri&#xE1;ngulos ...

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KEY TERMS AND CONCEPTS

TÉRMINOS Y CONCEPTOS CLAVE

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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