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

A contractor needs to estimate the amount of paint required to cover a specific part of a wall with a curved top edge in one hundred model homes. To do this accurately, the wall&#8217;s surface area must be calculated.If the curved edge follows a mathematical function, the problem reduces to finding the area under a given curve.To approximate this area, the region beneath the curve is divided into n number of rectangles, of width &#916;x. The sum of the areas of these rectangles provides an estimate of the total area.The height of each rectangle can be taken at the left endpoint or the right endpoint, which may lead to an overestimate or an underestimate depending on the curve&#8217;s shape.A more balanced estimate uses the function&#8217;s value at any point inside each subinterval, called the sample point.For each rectangle, the area is given by the function's value at the sample point multiplied by the width of the subinterval. Adding the areas of all the rectangles gives the approximate area.As the number of rectangles increases and their width decreases, the sum approaches the integral, which provides the exact area under the curve. This helps estimate the exact amount of paint required.

het oppervlakteprobleem

Het bepalen van de oppervlakte van een gebied met rechte randen is eenvoudig, aangezien geometrische formules voor rechthoeken, driehoeken en veelhoeken rechtstreeks kunnen worden toegepast. Traditionele geometrische methoden zijn echter ontoereikend wanneer een gebied een gebogen begrenzing heeft, zoals de oppervlakte onder een functie.<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>van<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>Het oppervlakteprobleem houdt in dat er een systematische methode wordt gevonden om dergelijke gebieden te meten. Een gangbare manier om dit probleem aan te pakken is door middel van benadering. In plaats van te trachten de oppervlakte aanvankelijk exact te berekenen, wordt het gebied onder de kromme eerst verdeeld in kleinere, eenvoudigere vormen. Een veelgebruikte methode is het benaderen van de oppervlakte met behulp van rechthoeken. Door de oppervlakten van deze rechthoeken te sommeren, verkrijgen we een schatting van de totale oppervlakte. De hoogte van elke rechthoek wordt bepaald door de functie te evalueren op specifieke punten binnen het interval. Verschillende keuzes voor deze punten kunnen leiden tot overschattingen of onderschattingen van de werkelijke oppervlakte.Naarmate het aantal rechthoeken toeneemt en hun breedtes kleiner worden, wordt de benadering nauwkeuriger. In de limiet, wanneer de breedte van elke rechthoek tot nul nadert, convergeert de som van hun oppervlakten tot een exacte waarde, die de werkelijke oppervlakte onder de kromme voorstelt. Dit proces vormt een rigoureuze basis voor het defini&#xEB;ren van oppervlakten in situaties waarin sprake is van gebogen begrenzingen.De methode waarbij gebogen gebieden worden benaderd door deze op te splitsen in eenvoudigere geometrische vormen reikt verder dan de wiskunde alleen en wordt op grote schaal toegepast in de natuurkunde, de economie en de techniek. Deze methode maakt nauwkeurige berekeningen mogelijk in situaties met geaccumuleerde grootheden, zoals de arbeid die wordt verricht door een vari&#xEB;rende kracht of de totale omzet over een bepaalde tijdsperiode.

Het oppervlakteprobleem

Het bepalen van de oppervlakte van een gebied met rechte randen is eenvoudig, aangezien geometrische formules voor rechthoeken, driehoeken en veelhoeken ...

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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...

distance problem

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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...

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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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 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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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...

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Finding Water Volume with Integration

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Work Done on a Nonlinear Spring

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substitution rule applied to definite integrals

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Using Substitution to Evaluate Definite Integrals

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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 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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