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

Die Mittelpunktregel hilft, die Fläche unter einer Kurve abzuschätzen, indem das Intervall in gleiche Teilintervalle unterteilt wird.Der Mittelpunkt jedes Teilintervalls wird als Mittelwert seiner beiden Endpunkte berechnet. Der Wert der Funktion an diesen Mittelpunkten gibt die Höhe der über ihnen gezeichneten Rechtecke an. Dieser Ansatz liefert oft eine genauere Schätzung als Methoden mit linken oder rechten Endpunkten, da er Fehler reduziert, die durch Unter- oder Überschätzung des Bereichs entstehen.Betrachten Sie eine Region neben einem Fluss. Es wird an den Seiten von zwei vertikalen Linien begrenzt, darunter eine horizontale Linie und ein geschwungenes Flussufer oben.Um seine Fläche zu schätzen, wird das Gebiet in mehrere Rechtecke gleicher Breite unterteilt.Die Höhe jedes Rechtecks wird mittels der Funktion in der Mitte seiner Basis bestimmt.Die Fläche jedes Rechtecks wird ermittelt, indem seine Höhe mit der Teilintervallbreite multipliziert wird. Diese Flächen werden addiert, um die Gesamtfläche unter der Kurve zu approximieren, was die Mittelpunktregelformel ergibt.Wenn die Anzahl der Rechtecke zunimmt und sie schmaler werden, verbessert sich die Schätzung und nähert sich der tatsächlichen Fläche.

mittelpunktregel

Die Approximation von Fl&#xE4;chen unter gekr&#xFC;mmten Begrenzungen ist ein h&#xE4;ufiges Problem in der angewandten Mathematik, insbesondere wenn eine exakte Berechnung schwierig oder unpraktisch ist. Eine bew&#xE4;hrte numerische Methode hierf&#xFC;r ist die Mittelpunktregel, die eine Sch&#xE4;tzung der Fl&#xE4;che unter einer Kurve durch rechteckige Approximationen &#xFC;ber ein vorgegebenes Intervall liefert.
Beschreibung der Mittelpunktregel
Die Mittelpunktregel beginnt mit der Unterteilung des gegebenen Intervalls in mehrere gleich gro&#xDF;e Teilintervalle. F&#xFC;r jedes Teilintervall wird der Mittelpunkt durch Mittelung der beiden Intervallendpunkte ermittelt. Der Funktionswert an diesem Mittelpunkt bestimmt die H&#xF6;he eines auf dem Teilintervall konstruierten Rechtecks. Da der Mittelpunkt h&#xE4;ufig n&#xE4;her am mittleren Funktionsverlauf innerhalb des Intervalls liegt, verringert diese Methode in der Regel den Fehler, der bei der Verwendung der linken oder rechten Endpunkte entstehen kann.
Geometrische Interpretation
Ein praktisches Beispiel f&#xFC;r die Anwendung der Mittelpunktregel ist die Sch&#xE4;tzung der Fl&#xE4;che eines an einen Fluss angrenzenden Gebiets. Das Gebiet ist links und rechts durch vertikale Linien, unten durch eine horizontale Linie und oben durch ein gekr&#xFC;mmtes Flussufer begrenzt. Da die obere Begrenzung unregelm&#xE4;&#xDF;ig ist, kann die Fl&#xE4;che nicht ohne Weiteres mit einfachen geometrischen Formeln berechnet werden.
Vorgehen zur Fl&#xE4;chenapproximation
Zur Ann&#xE4;herung der Fl&#xE4;che wird das Gebiet in mehrere gleich breite Rechtecke unterteilt. Die H&#xF6;he jedes Rechtecks wird durch Auswertung der Begrenzungskurve im Mittelpunkt der Rechteckbasis bestimmt. Die Fl&#xE4;che jedes Rechtecks ergibt sich aus dem Produkt von H&#xF6;he und Breite. Die Summe der Fl&#xE4;chen aller Rechtecke liefert eine Sch&#xE4;tzung der Gesamtfl&#xE4;che unter der Kurve, was dem wesentlichen Gedanken der Mittelpunktregel entspricht.
Genauigkeit der Methode
Mit zunehmender Anzahl der Rechtecke und abnehmender Breite der Rechtecke n&#xE4;hert sich die rechteckige Approximation immer genauer der Kurvenform an. Dadurch verbessert sich die Fl&#xE4;chensch&#xE4;tzung und kommt der tats&#xE4;chlichen Fl&#xE4;che des Gebiets zunehmend n&#xE4;her.

Mittelpunktregel

Die Approximation von Fl&#xE4;chen unter gekr&#xFC;mmten Begrenzungen ist ein h&#xE4;ufiges Problem in der angewandten Mathematik, insbesondere wenn eine ...

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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.

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Estimating Area with the Midpoint Rule

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