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

Ein Auftragnehmer muss die Menge an Farbe schätzen, die erforderlich ist, um einen bestimmten Teil einer Wand mit einer gebogenen Oberkante in hundert Modellhäusern zu bedecken. Um dies genau zu tun, muss die Oberfläche der Wand berechnet werden.Wenn die gekrümmte Kante einer mathematischen Funktion folgt, reduziert sich das Problem auf die Fläche unter einer gegebenen Kurve.Um diese Fläche zu approximieren, wird der Bereich unter der Kurve in n Rechtecke mit der Breite Δx unterteilt. Die Summe der Flächen dieser Rechtecke liefert eine Schätzung der Gesamtfläche.Die Höhe jedes Rechtecks kann am linken oder rechten Endpunkt gemessen werden, was je nach Form der Kurve zu einer Über- oder Unterschätzung führen kann.Eine ausgewogenere Schätzung verwendet den Wert der Funktion an jedem beliebigen Punkt innerhalb jedes Teilintervalls, dem sogenannten Stichprobenpunkt.Für jedes Rechteck wird die Fläche durch den Wert der Funktion am Abtastpunkt multipliziert mit der Breite des Teilintervalls gegeben. Addiert man die Flächen aller Rechtecke, erhält man die ungefähre Fläche.Wenn die Anzahl der Rechtecke zunimmt und ihre Breite abnimmt, nähert sich die Summe dem Integral, das die exakte Fläche unter der Kurve liefert. Das hilft dabei, die genaue Menge an Farbe abzuschätzen.

das flächenproblem

Die Berechnung der Fl&#xE4;che von Bereichen mit geraden Kanten ist unkompliziert, da geometrische Formeln f&#xFC;r Rechtecke, Dreiecke und Polygone direkt angewendet werden k&#xF6;nnen. Traditionelle geometrische Methoden sind jedoch unzureichend, sobald ein Bereich eine gekr&#xFC;mmte Begrenzung aufweist, wie dies bei der Fl&#xE4;che unter einer Funktion der Fall ist.<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>von<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>Das Fl&#xE4;chenproblem besteht darin, eine systematische Methode zur Bestimmung solcher Fl&#xE4;chen zu entwickeln. Ein m&#xF6;glicher Ansatz ist die N&#xE4;herung. Anstatt die Fl&#xE4;che sofort exakt zu berechnen, wird der Bereich unter der Kurve zuerst in kleinere, einfachere Formen unterteilt. Eine h&#xE4;ufig genutzte Methode besteht darin, die Fl&#xE4;che mithilfe von Rechtecken zu n&#xE4;hern. Indem man die Fl&#xE4;chen dieser Rechtecke summiert, erh&#xE4;lt man eine N&#xE4;herung der Gesamtfl&#xE4;che. Die H&#xF6;he jedes Rechtecks wird durch die Auswertung der Funktion an bestimmten Punkten innerhalb des Intervalls bestimmt. Unterschiedliche Auswahlen dieser Punkte k&#xF6;nnen zu &#xDC;ber- oder Untersch&#xE4;tzungen der tats&#xE4;chlichen Fl&#xE4;che f&#xFC;hren.Mit zunehmender Anzahl der Rechtecke und abnehmender Breite der Rechtecke wird die N&#xE4;herung immer genauer. Im Grenzwert, wenn die Breite jedes Rechtecks gegen null strebt, konvergiert die Summe der Rechtecksfl&#xE4;chen zu einem Grenzwert, der die tats&#xE4;chliche Fl&#xE4;che unter der Kurve darstellt. Dieser Prozess bietet eine rigorose Grundlage zur Definition von Fl&#xE4;chen mit gekr&#xFC;mmten Begrenzungen.Die Methode der N&#xE4;herung gekr&#xFC;mmter Bereiche durch Zerlegung in einfachere geometrische Formen geht &#xFC;ber die Mathematik hinaus und findet breite Anwendung in Physik, Wirtschaft und Ingenieurwesen. Sie erm&#xF6;glicht pr&#xE4;zise Berechnungen in Szenarien mit aufsummierten Gr&#xF6;&#xDF;en, wie etwa der von einer variierenden Kraft verrichteten Arbeit oder dem Gesamterl&#xF6;s &#xFC;ber einen bestimmten Zeitraum.

Das Flächenproblem

Die Berechnung der Fl&#xE4;che von Bereichen mit geraden Kanten ist unkompliziert, da geometrische Formeln f&#xFC;r Rechtecke, Dreiecke und Polygone ...

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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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