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

Rozważmy funkcję nad domkniętym przedziałem. Aby przybliżyć powierzchnię pod krzywą, podziel przedział na równe podprzedziały, tworząc prostokąty o tej samej szerokości.Wybierz dowolny punkt w podprzedziale i użyj wartości funkcji jako wysokości prostokąta. Mnożenie wysokości przez szerokość daje pole tego prostokąta. Całkowita powierzchnia pod krzywą jest przybliżona przez sumowanie pól wszystkich prostokątów i nazywana jest sumą Riemanna.Określona całka funkcji na danym przedziale definiowana jest jako granica jej sumy Riemanna, gdy liczba podprzedziałów zbliża się do nieskończoności, a szerokość każdego prostokąta staje się nieskończenie mała, pod warunkiem istnienia tego ograniczenia.Na przykład rozważmy samochód poruszający się z zmienną prędkością. Podziel całkowity czas podróży na równe przedziały czasowe.W każdym przedziale wybierz dowolny punkt czasowy i przyjmij odpowiadającą prędkość. Pomnóż prędkość przez szerokość przedziału, aby oszacować pokonany dystans.Suma tych elementów tworzy sumę Riemanna, a w tym przykładzie pole pod krzywą jest fizycznie interpretowane jako odległość pokonaną przez samochód. Gdy prędkość pozostaje dodatnia, całkowita odległość jest wyznaczana przez całkowanie funkcji prędkości w granicach.

całka oznaczona

Rozwa&#x17C;my funkcj&#x119; rzeczywist&#x105; zdefiniowan&#x105; na domkni&#x119;tym przedziale. Jednym z podstawowych cel&#xF3;w rachunku r&#xF3;&#x17C;niczkowego i ca&#x142;kowego jest wyznaczenie pola pod wykresem takiej funkcji. Gdy dok&#x142;adne obliczenia nie s&#x105; bezpo&#x15B;rednio mo&#x17C;liwe, pole to mo&#x17C;na oszacowa&#x107;, dziel&#x105;c przedzia&#x142; na sko&#x144;czon&#x105; liczb&#x119; r&#xF3;wnych podprzedzia&#x142;&#xF3;w. Ka&#x17C;dy podprzedzia&#x142; odpowiada prostok&#x105;towi, kt&#xF3;rego szeroko&#x15B;&#x107; jest r&#xF3;wna d&#x142;ugo&#x15B;ci podprzedzia&#x142;u, a wysoko&#x15B;&#x107; jest okre&#x15B;lona przez warto&#x15B;&#x107; funkcji w wybranym punkcie tego podprzedzia&#x142;u.
Sumy Riemanna i przybli&#x17C;enie pola
Dla ka&#x17C;dego podprzedzia&#x142;u wybiera si&#x119; dowolny punkt, a warto&#x15B;&#x107; funkcji w tym punkcie jest u&#x17C;ywana jako wysoko&#x15B;&#x107; prostok&#x105;ta. Pole ka&#x17C;dego prostok&#x105;ta uzyskuje si&#x119; przez pomno&#x17C;enie jego wysoko&#x15B;ci przez jego szeroko&#x15B;&#x107;. Zsumowanie p&#xF3;l wszystkich prostok&#x105;t&#xF3;w daje przybli&#x17C;enie ca&#x142;kowitego pola pod krzyw&#x105;. To przybli&#x17C;enie jest znane jako suma Riemanna. Wraz ze wzrostem liczby podprzedzia&#x142;&#xF3;w i zmniejszaniem ich szeroko&#x15B;ci przybli&#x17C;enie staje si&#x119; na og&#xF3;&#x142; dok&#x142;adniejsze, coraz lepiej odzwierciedlaj&#x105;c kszta&#x142;t krzywej.
Ca&#x142;ka oznaczona jako proces graniczny
Ca&#x142;ka oznaczona precyzuje ide&#x119; sumy Riemanna, rozwa&#x17C;aj&#x105;c graniczny przypadek, w kt&#xF3;rym podzia&#x142; przedzia&#x142;u staje si&#x119; niesko&#x144;czenie drobny. Ca&#x142;ka oznaczona funkcji na danym przedziale jest definiowana jako warto&#x15B;&#x107;, do kt&#xF3;rej zbli&#x17C;aj&#x105; si&#x119; sumy Riemanna, gdy liczba prostok&#x105;t&#xF3;w ro&#x15B;nie bez ogranicze&#x144;, a szeroko&#x15B;&#x107; ka&#x17C;dego prostok&#x105;ta staje si&#x119; pomijalnie ma&#x142;a, o ile taka warto&#x15B;&#x107; graniczna istnieje. W tym sensie ca&#x142;ka oznaczona stanowi dok&#x142;adne pole pod wykresem na rozwa&#x17C;anym przedziale.
Interpretacja w kontekstach fizycznych
Praktyczna ilustracja tej koncepcji pojawia si&#x119; w badaniach ruchu. Rozwa&#x17C;my samoch&#xF3;d poruszaj&#x105;cy si&#x119; po prostej trasie z pr&#x119;dko&#x15B;ci&#x105; zmieniaj&#x105;c&#x105; si&#x119; w czasie. Podzielenie ca&#x142;kowitego czasu podr&#xF3;&#x17C;y na r&#xF3;wne odcinki i wykorzystanie pr&#x119;dko&#x15B;ci w wybranym momencie w ka&#x17C;dym odcinku pozwala oszacowa&#x107; odleg&#x142;o&#x15B;&#x107; przebyt&#x105; na tym odcinku. Zsumowanie tych odleg&#x142;o&#x15B;ci daje sum&#x119; Riemanna, kt&#xF3;ra przybli&#x17C;a ca&#x142;kowit&#x105; przebyt&#x105; drog&#x119;. Gdy pr&#x119;dko&#x15B;&#x107; pozostaje dodatnia w ca&#x142;ym przedziale, ca&#x142;ka oznaczona funkcji pr&#x119;dko&#x15B;ci odpowiada ca&#x142;kowitej drodze przebytej przez samoch&#xF3;d.

Całka oznaczona

Rozwa&#x17C;my funkcj&#x119; rzeczywist&#x105; zdefiniowan&#x105; na domkni&#x119;tym przedziale. Jednym z podstawowych cel&#xF3;w rachunku ...

Education

JoVE Core

Calculus

Rachunek różniczkowy i całkowy

Unit

Jednostka

Integrals

Całki

KEY TERMS AND CONCEPTS

KLUCZOWE TERMINY I POJĘCIA

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

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...

distance problem

Estimating Distance from Changing Velocity

Estimating distance from changing velocity uses small time intervals and velocity values. When an object speeds up or slows down, its total displacement can be found by adding many short pieces of motion. This method gives an approximation first, and integral calculus gives the exact result later. A runner who accelerates during the first three seconds of a race shows how the idea works. Velocity can be measured every half-second and multiplied by the 0.5-second time step. Using left-endpoint...

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...

sums of power

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...

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...

properties of definite integral i

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...

properties of definite integral iii

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...

fundamental theorem of calculus i

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...

fundamental theorem of calculus i: problem solving

fundamental-theorem-of-calculus-i-problem-solving

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...

fundamental theorem of calculus ii

Definite Integrals with Antiderivatives

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...

application of integration: problem solving

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Integration in Breathing Cycle Modeling

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...

indefinite integrals

Finding Water Volume with Integration

Integration can be used to find the total volume of water in a tank when the inflow rate changes over time. In this example, the pump starts at 5 L/min. The inflow rate then increases by 2 L/min for each additional minute. That changing rate can be written as a linear function. To measure how much water has been added, the inflow rate function is integrated over time. The total water volume V(t) comes from this integral. When the power rule is used, the result includes both a quadratic term...

net change theorem

Calculus for Reservoir Overflow Prediction

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...

substitution rule applied to indefinite integrals

substitution-rule-applied-to-indefinite-integrals

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,...

substitution rule applied to definite integrals

substitution-rule-applied-to-definite-integrals

Using Substitution to Evaluate Definite Integrals

Substitution makes definite integrals easier to evaluate when the integrand is a composite function. It works by reversing the chain rule from differentiation. When the expression includes an inner function and its derivative, substitution can simplify the setup quickly. The method starts by choosing the inner function and naming it with a new variable. Then its differential is found and used to replace the original variable terms. The limits of integration are also changed by plugging the...

growth models with integration: problem solving

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Finding Fish Biomass with Integrals

Fish biomass can be modeled with integration when the growth rate is known over time. In ecology, biomass means the total weight of a fish population in a body of water. If the growth rate of that biomass is given as a function of time, calculus can be used to find the total biomass at a later date. The growth rate is written as G(t), where t is time in years and G(t) is the rate of change of biomass in kilograms per year. The biomass itself is written as B(t). Because the growth rate shows...

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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