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

Bir müteahhit, yüz model evde duvarın belirli bir bölümünü kavisli üst kenarla kaplamak için gereken boya miktarını tahmin etmelidir. Bunu doğru yapmak için duvarın yüzey alanı hesaplanmalıdır.Eğer eğri kenar matematiksel bir fonksiyonu takip ederse, problem verilen bir eğrinin altındaki alanı bulmaya indirgenir.Bu alanı yaklaşık olarak belirlemek için, eğrinin altındaki bölge, genişliği Δx olan n sayıda dikdörtgene bölünür. Bu dikdörtgenlerin alanlarının toplamı, toplam alanın bir tahminini sağlar.Her dikdörtgenin yüksekliği sol uç noktada veya sağ uçta alınabilir; bu da eğrinin şekline bağlı olarak fazla veya az tahmin yapılmasına yol açabilir.Daha dengeli bir tahmin, fonksiyonun değerini her alt aralık içindeki herhangi bir noktada kullanır; örnek noktası olarak adlandırılır.Her dikdörtgen için, alan örnek noktasındaki fonksiyonun değerinin alt aralık genişliğiyle çarpılması ile verilir. Tüm dikdörtgenlerin alanlarını eklediğinizde yaklaşık alan elde edilir.Dikdörtgenlerin sayısı arttıkça ve genişlikleri azaldıkça, toplam eğrinin altındaki tam alanı sağlayan integrale yaklaşır. Bu, gereken tam boya miktarını tahmin etmeye yardımcı olur.

alan problemi

D&#xFC;z kenarlara sahip bir b&#xF6;lgenin alan&#x131;n&#x131; belirlemek g&#xF6;rece kolayd&#x131;r; &#xE7;&#xFC;nk&#xFC; dikd&#xF6;rtgenler, &#xFC;&#xE7;genler ve &#xE7;okgenler i&#xE7;in ge&#xE7;erli olan geometrik form&#xFC;ller do&#x11F;rudan uygulanabilir. Ancak bir b&#xF6;lgenin s&#x131;n&#x131;r&#x131; e&#x11F;ri oldu&#x11F;unda&#x2014;&#xF6;rne&#x11F;in bir fonksiyonun alt&#x131;ndaki alan s&#xF6;z konusu oldu&#x11F;unda&#x2014;geleneksel geometrik y&#xF6;ntemler yetersiz kal&#x131;r.<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>&#x130;tibaren<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>Alan problemi, bu t&#xFC;r b&#xF6;lgelerin &#xF6;l&#xE7;&#xFC;lmesi i&#xE7;in sistematik bir y&#xF6;ntem bulmay&#x131; ama&#xE7;lar. Bu problemi &#xE7;&#xF6;zmeye y&#xF6;nelik yakla&#x15F;&#x131;mlardan biri yakla&#x15F;t&#x131;rma yoluyla yap&#x131;lan bir yakla&#x15F;&#x131;md&#x131;r. Alan&#x131; ba&#x15F;lang&#x131;&#xE7;ta tam olarak hesaplamaya &#xE7;al&#x131;&#x15F;mak yerine, e&#x11F;rinin alt&#x131;ndaki b&#xF6;lge &#xF6;nce daha k&#xFC;&#xE7;&#xFC;k ve daha basit &#x15F;ekillere ayr&#x131;l&#x131;r. Yayg&#x131;n olarak kullan&#x131;lan bir y&#xF6;ntem, alan&#x131;n dikd&#xF6;rtgenler kullan&#x131;larak yakla&#x15F;t&#x131;r&#x131;lmas&#x131;d&#x131;r. Bu dikd&#xF6;rtgenlerin alanlar&#x131; topland&#x131;&#x11F;&#x131;nda, toplam alana ili&#x15F;kin bir tahmin elde edilir. Her bir dikd&#xF6;rtgenin y&#xFC;ksekli&#x11F;i, aral&#x131;k &#xFC;zerindeki belirli noktalarda fonksiyonun de&#x11F;erlendirilmesiyle belirlenir. Bu noktalar&#x131;n farkl&#x131; &#x15F;ekillerde se&#xE7;ilmesi, ger&#xE7;ek alan&#x131;n oldu&#x11F;undan b&#xFC;y&#xFC;k ya da k&#xFC;&#xE7;&#xFC;k tahmin edilmesine yol a&#xE7;abilir.Dikd&#xF6;rtgenlerin say&#x131;s&#x131; artt&#x131;k&#xE7;a ve geni&#x15F;likleri k&#xFC;&#xE7;&#xFC;ld&#xFC;k&#xE7;e, yakla&#x15F;t&#x131;rma giderek daha hassas h&#xE2;le gelir. S&#x131;n&#x131;r durumda, her bir dikd&#xF6;rtgenin geni&#x15F;li&#x11F;i s&#x131;f&#x131;ra yakla&#x15F;t&#x131;&#x11F;&#x131;nda, alanlar&#x131;n&#x131;n toplam&#x131; e&#x11F;rinin alt&#x131;ndaki ger&#xE7;ek alan&#x131; temsil eden kesin bir de&#x11F;ere yak&#x131;nsar. Bu s&#xFC;re&#xE7;, e&#x11F;ri s&#x131;n&#x131;rlara sahip b&#xF6;lgeler s&#xF6;z konusu oldu&#x11F;unda alan kavram&#x131;n&#x131;n tan&#x131;mlanmas&#x131; i&#xE7;in sa&#x11F;lam ve kesin bir temel sunar.E&#x11F;ri b&#xF6;lgelerin daha basit geometrik &#x15F;ekillere ayr&#x131;larak yakla&#x15F;t&#x131;r&#x131;lmas&#x131; y&#xF6;ntemi yaln&#x131;zca matematikle s&#x131;n&#x131;rl&#x131; de&#x11F;ildir; fizik, ekonomi ve m&#xFC;hendislik gibi alanlarda da yayg&#x131;n bi&#xE7;imde uygulan&#x131;r. Bu y&#xF6;ntem, de&#x11F;i&#x15F;ken bir kuvvetin yapt&#x131;&#x11F;&#x131; i&#x15F; ya da zaman i&#xE7;indeki toplam gelir gibi birikimli niceliklerin hesaplanmas&#x131;n&#x131; gerektiren durumlarda hassas hesaplamalar yap&#x131;lmas&#x131;na olanak tan&#x131;r.

Alan Problemi

D&#xFC;z kenarlara sahip bir b&#xF6;lgenin alan&#x131;n&#x131; belirlemek g&#xF6;rece kolayd&#x131;r; &#xE7;&#xFC;nk&#xFC; dikd&#xF6;rtgenler, ...

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Calculus

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Integrals

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KEY TERMS AND CONCEPTS

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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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Riemann Sums and Area Estimates

Riemann sums and area estimates are a key part of calculus for finding the area under a curve. When a real-valued function is defined on a closed interval, the area under its graph can be approximated if an exact calculation is not easy to do. The interval is split into equal subintervals. Each subinterval becomes a rectangle. The rectangle’s width is the length of the subinterval, and its height comes from the function value at one chosen point inside that part of the interval. Multiplying...

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Gauss Sums for Riemann Rectangles

Riemann sums use rectangles to estimate the area under a curve in definite integration. The interval is split into subintervals, and the rectangle areas are added together. When those rectangle heights follow a regular pattern, sum formulas can make the calculation faster and more exact. This page focuses on sums that appear in arithmetic and polynomial sequences. The sum of consecutive integers, squares, and cubes is especially useful when the partition is uniform or when the data has a...

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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...

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Definite Integrals for Car Motion

Definite integrals help describe a car’s motion over time by linking velocity to displacement. In a velocity-time graph, the definite integral shows how velocity builds up over a chosen time interval. The area under the curve gives the total displacement, so the graph becomes a clear picture of how far the car moves. One key idea is linearity of definite integrals. This property lets you add or subtract velocity functions while keeping their accumulated effect. If a car’s motion is made of...

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Definite Integrals in Motion Analysis

Definite integrals can be used to find displacement from a velocity-time graph. A velocity function v(t) describes how motion changes over time. The definite integral of v(t) from a to b gives the net displacement between those times. It also represents the signed area under the velocity curve between the two points. Two useful properties make definite integrals easier to work with in motion problems. The additivity property says that if a continuous function is integrated over adjacent...

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The definite integral helps describe motion by showing how far an object travels over time. In this lesson, the key ideas are the Positivity Property and the Comparison Property of definite integrals. Both ideas connect integration to velocity and displacement in a way that is easy to interpret in real situations. The Positivity Property says that if an object’s velocity is nonnegative, then its total displacement over the time interval is also nonnegative. Nonnegative velocity means the...

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Definite Integrals: A Step-by-Step Strategy

Definite integrals are solved best with a clear, step-by-step strategy. The first step is to identify what the problem is asking for. The quantity may be accumulation, area, force, or probability. It is also important to note the integration limits and any constraints that affect the setup. Before evaluating the integral, look for properties that can simplify the work. If the expression has multiple parts, break it into smaller integrals. Known values from earlier work may help too. Symmetry...

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The Fundamental Theorem of Calculus connects accumulation and rate of change. In this lesson, the idea is shown with a water tank that is filled by a pump. The pump rate changes over time, so the total water collected must be found by adding up the changing inflow. In water management, this kind of calculation matters for pressure control, operation timing, and system safety. The flow rate is given by a time-dependent function. To find how much water has entered the tank from the start to a...

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The Fundamental Theorem of Calculus, Part 2, connects definite integrals with antiderivatives. It shows that the area under a continuous curve can be found without adding infinitely many tiny rectangles one by one. Instead, the integral can be evaluated directly from an antiderivative. If a function f(x) is continuous on a closed interval [a, b], then its definite integral over that interval can be computed using any antiderivative F(x). The result is written as F(b) - F(a). This works because...

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Integration can be used to model the volume of air inhaled during breathing. The respiratory cycle is the repeating intake and expulsion of air, and it usually lasts about five seconds. By treating airflow as a function of time, students can see how the amount of air in the lungs changes during a breath. Airflow during respiration is not constant. It changes in a smooth, wave-like pattern, so a sinusoidal function is a good model. In a normal breathing cycle, the airflow rate reaches a maximum...

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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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