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.

area problem

Determining the area of a region with straight edges is straightforward, as geometric formulas for rectangles, triangles, and polygons can be applied directly. However, traditional geometric methods are insufficient when a region has a curved boundary, such as the area under a function.<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:30.21%;"><img style="aspect-ratio:300/24;" src="https://cloudfront.jove.com/files/ftp_upload/19973/19973_PT_Equation_1_editor_19973.svg" alt="Mathematical function y=f(x), equation, calculus, algebra, static diagram." data-altupdatedat="1775223722000" data-alt="19973_PT_Equation_1.svg" /></figure>from<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:20.57%;"><img style="aspect-ratio:300/24;" src="https://cloudfront.jove.com/files/ftp_upload/19973/19973_PT_Equation_2_editor_19973.svg" alt="Equation showing a variable range: x=a to x=b." data-altupdatedat="1775223722000" data-alt="19973_PT_Equation_2.svg" /></figure>The area problem involves finding a systematic way to measure such regions. One approach to solving this problem is through approximation. Instead of attempting to compute the area exactly at the outset, the region under the curve is first divided into smaller, simpler shapes. A common method involves approximating the area using rectangles. By summing the areas of these rectangles, we obtain an estimate of the total area. The height of each rectangle is determined by evaluating the function at specific points along the interval. Different choices for these points may lead to overestimations or underestimations of the actual area.As the number of rectangles increases and their widths become smaller, the approximation becomes more precise. In the limit, as the width of each rectangle approaches the minimum value, the sum of their areas converges to an exact value, which represents the true area under the curve. This process provides a rigorous foundation for defining areas in cases where curved boundaries are involved.The method of approximating curved regions by breaking them into simpler geometric shapes extends beyond mathematics and is widely applied in physics, economics, and engineering. It allows for precise computations in scenarios involving accumulated quantities, such as work done by a varying force or total revenue over time.

Area Problem

Determining the area of a region with straight edges is straightforward, as geometric formulas for rectangles, triangles, and polygons can be applied ...

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JoVE Core

Calculus

Unit

Integrals

KEY TERMS AND CONCEPTS

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

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

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

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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 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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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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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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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...

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Work Done on a Nonlinear Spring

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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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