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

La regla del punto medio ayuda a estimar el área bajo una curva dividiendo el intervalo en subintervalos iguales.El punto medio de cada subintervalo se calcula como la media de sus dos extremos. El valor de la función en estos puntos medios da la altura de los rectángulos dibujados sobre ellos. Este enfoque suele proporcionar una estimación más precisa que los que utilizan extremos izquierdo o derecho, ya que reduce los errores causados por subestimar o sobreestimar el área.Considera una región junto a un río. Está delimitado por dos líneas verticales en los laterales, una línea horizontal por debajo y una orilla curva del río arriba.Para estimar su superficie, la región se divide en varios rectángulos de igual anchura.La altura de cada rectángulo se determina usando la función en el punto medio de su base.El área de cada rectángulo se determina multiplicando su altura por la anchura del subintervalo. Estas áreas se suman para aproximar el área total bajo la curva, dando la fórmula de la regla del punto medio.A medida que aumenta el número de rectángulos y se estrechan, la estimación mejora y se acerca al área real.

regla del punto medio

Aproximar &#xE1;reas bajo curvas delimitadas es un problema com&#xFA;n en matem&#xE1;ticas aplicadas, especialmente cuando un c&#xE1;lculo exacto resulta dif&#xED;cil o poco pr&#xE1;ctico. Un m&#xE9;todo num&#xE9;rico eficaz para este prop&#xF3;sito es la Regla del Punto Medio, que proporciona una estimaci&#xF3;n del &#xE1;rea bajo una curva mediante aproximaciones rectangulares en un intervalo espec&#xED;fico.
Descripci&#xF3;n de la regla del punto medio
La Regla del Punto Medio comienza dividiendo el intervalo dado en varios subintervalos iguales. Para cada subintervalo, el punto medio se identifica promediando los valores en sus dos extremos. El valor de la funci&#xF3;n en este punto medio se utiliza para determinar la altura de un rect&#xE1;ngulo sobre el subintervalo. Dado que el punto medio suele estar m&#xE1;s cerca del comportamiento promedio de la funci&#xF3;n en ese intervalo, este m&#xE9;todo tiende a reducir el error asociado con la sobreestimaci&#xF3;n o subestimaci&#xF3;n al utilizar los extremos izquierdo o derecho.
Interpretaci&#xF3;n geom&#xE9;trica
Un ejemplo pr&#xE1;ctico de la Regla del Punto Medio puede ilustrarse estimando el &#xE1;rea de una regi&#xF3;n adyacente a un r&#xED;o. La regi&#xF3;n est&#xE1; delimitada a la izquierda y a la derecha por l&#xED;neas verticales, por debajo por una l&#xED;nea horizontal y por encima, por una orilla curva. Dado que el l&#xED;mite superior es irregular, su &#xE1;rea no se puede calcular f&#xE1;cilmente mediante f&#xF3;rmulas geom&#xE9;tricas simples.
Proceso de aproximaci&#xF3;n del &#xE1;rea
Para aproximar el &#xE1;rea de esta regi&#xF3;n, se divide en varios rect&#xE1;ngulos de igual ancho. La altura de cada rect&#xE1;ngulo se determina evaluando la curva del l&#xED;mite en el punto medio de la base del rect&#xE1;ngulo. El &#xE1;rea de cada rect&#xE1;ngulo se calcula multiplicando su altura por su ancho. La suma de las &#xE1;reas de todos los rect&#xE1;ngulos produce una estimaci&#xF3;n del &#xE1;rea total bajo la curva, lo que constituye la esencia de la Regla del Punto Medio.
Precisi&#xF3;n del m&#xE9;todo
A medida que aumenta el n&#xFA;mero de rect&#xE1;ngulos y disminuye su ancho, la aproximaci&#xF3;n rectangular se ajusta m&#xE1;s a la forma de la curva. En consecuencia, el &#xE1;rea estimada mejora y se aproxima al &#xE1;rea real de la regi&#xF3;n.

Regla del Punto Medio

Aproximar &#xE1;reas bajo curvas delimitadas es un problema com&#xFA;n en matem&#xE1;ticas aplicadas, especialmente cuando un c&#xE1;lculo exacto resulta ...

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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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