Name: estimating area with the midpoint rule
Uploaded: 2026-01-12T12:24:26.000Z
Duration: 1 min 20 s
Channel: JoVE

The Midpoint rule helps estimate the area under a curve by dividing the interval into equal subintervals.The midpoint of each subinterval is calculated as the average of its two endpoints. The function&#8217;s value at these midpoints gives the height of rectangles drawn over them. This approach often gives a more accurate estimate than methods using left or right endpoints, as it reduces errors caused by underestimating or overestimating the area.Consider a region beside a river. It is bounded by two vertical lines on the sides, a horizontal line below, and a curved riverbank above.To estimate its area, the region is divided into several rectangles of equal width.The height of each rectangle is found using the function at the midpoint of its base.The area of each rectangle is found by multiplying its height by the subinterval width. These areas are added to approximate the total area under the curve, giving the Midpoint rule formula.As the number of rectangles increases and they become narrower, the estimate improves and approaches the actual area.

midpoint rule

Approximating areas under curved boundaries is a common problem in applied mathematics, particularly when an exact calculation is difficult or impractical. One effective numerical method for this purpose is the Midpoint Rule, which provides an estimate of the area under a curve by using rectangular approximations over a specified interval.<h3>Description of the Midpoint Rule</h3>The Midpoint Rule begins by dividing the given interval into a number of equal subintervals. For each subinterval, the midpoint is identified by averaging the values at its two endpoints. The value of the function at this midpoint is then used to determine the height of a rectangle constructed over the subinterval. Because the midpoint often lies closer to the average behavior of the function on that interval, this method tends to reduce the error associated with overestimation or underestimation that can occur when left or right endpoints are used.<h3>Geometric Interpretation</h3>A practical example of the Midpoint Rule can be illustrated by estimating the area of a region adjacent to a river. The region is bounded on the left and right by vertical lines, below by a horizontal line, and above by a curved riverbank. Since the upper boundary is irregular, its area cannot be easily computed using simple geometric formulas.<h3>Area Approximation Process</h3>To approximate the area of this region, it is divided into several rectangles of equal width. The height of each rectangle is determined by evaluating the boundary curve at the midpoint of the rectangle&#8217;s base. The area of each rectangle is calculated by multiplying its height by its width. Adding the areas of all rectangles produces an estimate of the total area under the curve, which is the essence of the Midpoint Rule.<h3>Accuracy of the Method</h3>As the number of rectangles increases and their widths decrease, the rectangular approximation more closely follows the shape of the curve. Consequently, the estimated area improves and approaches the true area of the region.

Midpoint Rule

Approximating areas under curved boundaries is a common problem in applied mathematics, particularly when an exact calculation is difficult or ...

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Integrals

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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 subinterval.

This midpoint choice often matches the average behavior of the curve better than using the left or right endpoint. That can reduce the error that may happen with overestimation or underestimation. In this way, the method gives a closer area estimate than simple endpoint-based rectangles in many cases.

The idea can be shown with a region next to a river. The region is bounded on the left and right by vertical lines, below by a horizontal line, and above by a curved riverbank. Since the upper edge is irregular, its area is not easy to find with simple geometry.

The area is approximated by dividing the region into several rectangles of equal width. Each rectangle’s area is found by multiplying its width by its midpoint height. Adding the areas of all the rectangles gives the estimated total area under the curve.

The estimate improves as the number of rectangles increases and their widths get smaller. The rectangles then follow the curve more closely. As a result, the approximation approaches the true area of the region.

Estimating Area with the Midpoint Rule

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key terms and concepts

Learn how the Midpoint Rule estimates area under a curve with equal-width rectangles, midpoint heights, and better accuracy as intervals get smaller.

Estimating Area with the Midpoint Rule | 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

Estimating Area with the Midpoint Rule

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