Name: riemann sums and area estimates
Uploaded: 2026-01-12T12:24:23.000Z
Duration: 1 min 29 s
Channel: JoVE

Kapalı bir aralık üzerinde bir fonksiyon ele alın. Eğrin altındaki alanı yaklaşık olarak belirlemek için, aralığı eşit alt aralıklara bölerek eşit genişlikte dikdörtgenler oluşturun.Alt aralık içinde rastgele bir nokta seçin ve oradaki fonksiyon değerini dikdörtgenin yüksekliği olarak kullanın. Yüksekliği genişlikle çarpmak, o dikdörtgenin alanını verir. Eğrinin altındaki toplam alan, tüm dikdörtgenlerin alanlarının toplamıyla yaklaşık olarak hesaplanır ve Riemann toplamı olarak adlandırılır.Belirli bir aralık üzerindeki belirli bir fonksiyonun integrali, alt aralıkların sayısı sonsuzluğa yaklaştıkça ve her dikdörtgenin genişliği sonsuz küçük boyutta küçük hale geldikçe, bu sınır mevcut olduğunda Riemann toplamının sınırı olarak tanımlanır.Örneğin, değişken hızda seyahat eden bir arabayı düşünün. Toplam seyahat süresini eşit zaman aralıklarına böl.Her aralıkta, rastgele bir zaman noktası seçer ve karşılık gelen hızı alır. Hızı aralık genişliğiyle çarparak kat edilen mesafeyi tahmin edin.Bunların toplamı bir Riemann toplamı oluşturur ve bu örnekte, eğrinin altındaki alan fiziksel olarak arabanın kat ettiği mesafe olarak yorumlanır. Hız pozitif kaldığında, toplam mesafe hız fonksiyonunun sınırlar içinde entegre edilmesiyle bulunur.

belirli integral

Kapal&#x131; bir aral&#x131;kta tan&#x131;ml&#x131; ger&#xE7;ek de&#x11F;erli bir fonksiyonu d&#xFC;&#x15F;&#xFC;nelim. Kalk&#xFC;l&#xFC;s&#xFC;n temel ama&#xE7;lar&#x131;ndan biri, bu t&#xFC;r bir fonksiyonun grafi&#x11F;inin alt&#x131;nda kalan alan&#x131; belirlemektir. Kesin bir hesaplaman&#x131;n kolayca yap&#x131;lamad&#x131;&#x11F;&#x131; durumlarda, bu alan aral&#x131;&#x11F;&#x131; sonlu say&#x131;da e&#x15F;it alt aral&#x131;&#x11F;a b&#xF6;lerek tahmin edilebilir. Her bir alt aral&#x131;k, geni&#x15F;li&#x11F;i alt aral&#x131;&#x11F;&#x131;n uzunlu&#x11F;una e&#x15F;it olan ve y&#xFC;ksekli&#x11F;i o alt aral&#x131;kta se&#xE7;ilen bir noktadaki fonksiyon de&#x11F;erine g&#xF6;re belirlenen bir dikd&#xF6;rtgene kar&#x15F;&#x131;l&#x131;k gelir.
Riemann Toplamlar&#x131; ve Alan Yakla&#x15F;&#x131;m&#x131;
Her alt aral&#x131;k i&#xE7;in keyfi bir nokta se&#xE7;ilir ve bu noktadaki fonksiyon de&#x11F;eri dikd&#xF6;rtgenin y&#xFC;ksekli&#x11F;i olarak al&#x131;n&#x131;r. Her bir dikd&#xF6;rtgenin alan&#x131;, y&#xFC;ksekli&#x11F;i ile geni&#x15F;li&#x11F;inin &#xE7;arp&#x131;m&#x131;yla elde edilir. T&#xFC;m dikd&#xF6;rtgenlerin alanlar&#x131;n&#x131;n toplanmas&#x131;, e&#x11F;rinin alt&#x131;nda kalan toplam alan&#x131;n yakla&#x15F;&#x131;k de&#x11F;erini verir. Bu yakla&#x15F;&#x131;ma Riemann toplam&#x131; denir. Alt aral&#x131;klar&#x131;n say&#x131;s&#x131; artt&#x131;k&#xE7;a ve her birinin geni&#x15F;li&#x11F;i azald&#x131;k&#xE7;a, bu yakla&#x15F;&#x131;m genellikle daha do&#x11F;ru hale gelir ve e&#x11F;rinin &#x15F;eklini daha iyi yans&#x131;t&#x131;r.
Limit S&#xFC;reci Olarak Belirli &#x130;ntegral
Belirli integral, aral&#x131;&#x11F;&#x131;n b&#xF6;l&#xFC;nmesinin sonsuz &#xF6;l&#xE7;&#xFC;de incelmesi durumunda ne oldu&#x11F;una bakarak Riemann toplam&#x131; fikrini daha ileri bir d&#xFC;zeye ta&#x15F;&#x131;r. Belirli bir aral&#x131;k &#xFC;zerinde tan&#x131;ml&#x131; bir fonksiyonun belirli integrali, dikd&#xF6;rtgen say&#x131;s&#x131; s&#x131;n&#x131;rs&#x131;z bi&#xE7;imde artarken ve her bir dikd&#xF6;rtgenin geni&#x15F;li&#x11F;i ihmal edilebilir derecede k&#xFC;&#xE7;&#xFC;l&#xFC;rken Riemann toplamlar&#x131;n&#x131;n yakla&#x15F;t&#x131;&#x11F;&#x131; de&#x11F;er olarak tan&#x131;mlan&#x131;r; bu limit mevcutsa. Bu anlamda belirli integral, belirtilen aral&#x131;kta e&#x11F;rinin alt&#x131;nda kalan kesin alan&#x131; temsil eder.
Fiziksel Ba&#x11F;lamlarda Yorum
Bu kavram&#x131;n pratik bir &#xF6;rne&#x11F;i, hareketin incelenmesinde ortaya &#xE7;&#x131;kar. Zamanla de&#x11F;i&#x15F;en bir h&#x131;zla do&#x11F;rusal bir yol boyunca ilerleyen bir arac&#x131; d&#xFC;&#x15F;&#xFC;nelim. Toplam yolculuk s&#xFC;resi e&#x15F;it zaman aral&#x131;klar&#x131;na b&#xF6;l&#xFC;nd&#xFC;&#x11F;&#xFC;nde ve her bir zaman aral&#x131;&#x11F;&#x131;nda se&#xE7;ilen bir andaki h&#x131;z de&#x11F;eri kullan&#x131;ld&#x131;&#x11F;&#x131;nda, o aral&#x131;kta kat edilen mesafe tahmin edilebilir. Bu mesafelerin toplanmas&#x131;, kat edilen toplam mesafeyi yakla&#x15F;t&#x131;ran bir Riemann toplam&#x131; olu&#x15F;turur. H&#x131;z&#x131;n t&#xFC;m aral&#x131;k boyunca pozitif kald&#x131;&#x11F;&#x131; durumlarda h&#x131;z fonksiyonunun belirli integrali, arac&#x131;n kat etti&#x11F;i toplam mesafeye kar&#x15F;&#x131;l&#x131;k gelir.

Belirli integral

Kapal&#x131; bir aral&#x131;kta tan&#x131;ml&#x131; ger&#xE7;ek de&#x11F;erli bir fonksiyonu d&#xFC;&#x15F;&#xFC;nelim. Kalk&#xFC;l&#xFC;s&#xFC;n temel ...

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Integrals

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Definite Integral

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 height by width gives the area of each rectangle.

Adding the areas of all the rectangles gives a Riemann sum. This sum is an approximation of the total area under the curve. As the number of subintervals increases and the widths get smaller, the estimate usually becomes more accurate because it follows the shape of the curve more closely.

The definite integral builds on this idea by taking the limit of the Riemann sums. It looks at what happens as the partition of the interval becomes infinitely fine and the rectangle widths become negligibly small, provided that the limit exists. In that case, the definite integral gives the exact area under the curve over the interval.

This idea also has a physical meaning in motion. A car moving along a straight path with changing velocity can be studied by dividing the travel time into equal parts. Using the velocity at a chosen time in each part gives an estimate for the distance traveled in that segment. Adding those distances gives a Riemann sum for total distance, and when the velocity stays positive, the definite integral of velocity over time matches the total distance covered.

Riemann Sums and Area Estimates

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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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definite integral

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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Estimating Area with the Midpoint Rule

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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...

properties of definite integral ii

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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Definite Integral Rules for Motion

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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Accumulation and Rates in the Fundamental Theorem

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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The Net Change Theorem links a function’s rate of change to its total accumulated change over an interval. In calculus, this means the definite integral of a derivative on [a,b] gives the net change in the original function. The theorem is useful in physics, economics, and engineering because it turns a rate into a total amount. One clear application is a reservoir that receives water from two inflow sources at the same time. The reservoir starts with 260 million cubic meters of water. Each...

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Work Done on a Nonlinear Spring

Work done on a nonlinear spring can be found by using integration and the substitution rule. A linear spring follows Hooke’s law, so its restoring force increases in direct proportion to displacement. That makes the force-displacement graph a straight line and the work calculation straightforward. A nonlinear spring does not obey Hooke’s law. Its restoring force changes with position in a non-proportional way, so the force-displacement graph is curved instead of linear. In the example shown,...

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Using Substitution to Evaluate Definite Integrals

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key terms and concepts

Learn how Riemann sums use rectangles to estimate area under a curve and how the definite integral gives the exact value as the intervals shrink.

Riemann Sums and Area Estimates | 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

Riemann Sums and Area Estimates

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