Name: vertical slices for area between curves
Uploaded: 2026-07-24T11:53:29.280Z
Duration: 1 min 21 s
Channel: JoVE

Consider two continuous functions on a closed interval [a,b]. The region &#x1D446; lies between the curves, bounded vertically by the functions and horizontally by a and b.The area between the curves can be estimated by dividing the region into n vertical strips of equal width. Each strip is treated as a rectangle, which is used in the Riemann sum method.Consider the ith strip. Take its base, and then the height of the strip at xi. The area of this rectangle is calculated by multiplying its height by its base.The total estimated area of this region is found by summing the areas of all n rectangles. These rectangles do not cover the full area exactly.But in the limit as n approaches infinity and the number of rectangles increases, the total area in the sum approaches the exact value. This limit is defined as the integral and yields the exact area between the curves.This concept is used in economics to find the inequality gap, which is the area between the line of perfect equality and the curve of actual income distribution.

area between curves: integrating with respect to x

Area Between Curves: Integrating With Respect to x

Consider two continuous functions defined on a closed interval from a to b. The region between these curves is bounded vertically by their graphs and horizontally by the endpoints of the interval. The objective is to measure the area of this region.An initial estimate of the area can be obtained by dividing the interval into a large number of narrow vertical strips of equal width. Each strip is approximated by a rectangle whose height is given by the vertical difference between the two functions at a selected point within the strip. Adding the areas of these rectangles produces a Riemann sum approximation of the region. This process is represented by<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:30.91%;"><img style="aspect-ratio:300/48;" src="https://cloudfront.jove.com/files/ftp_upload/40564/100562_Resized_Equation_1_editor_40564.svg" alt="100562_Resized_Equation_1.svg" /></figure>Because the rectangles do not exactly match the curved boundaries, this sum provides only an approximation when a finite number of strips is used. However, as the number of strips increases and their width decreases, the approximation improves. In the limiting case, the rectangles become infinitely thin, and the summation converges to the exact area.&#xA0;This limiting process replaces the Riemann sum with a definite integral, which computes the area using infinitely thin vertical slices. The exact area between the curves is given by<figure class="image image-style-align-center" style="display:table;margin-left:auto;margin-right:auto;"><img style="aspect-ratio:300/48;" src="https://cloudfront.jove.com/files/ftp_upload/40564/100562_Resized_Equation_2_editor_40564.svg" alt="100562_Resized_Equation_2.svg" /></figure>This method of finding the area between curves has important applications beyond pure mathematics. In economics, for example, the area between the line of perfect equality and the curve representing actual income distribution is used to quantify income inequality. The magnitude of this area provides a numerical measure of the inequality gap, demonstrating how geometric ideas from calculus can be applied to real-world economic analysis.

Consider two continuous functions defined on a closed interval from a to b. The region between these curves is bounded vertically by their graphs and ...

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area-between-curves-integrating-with-respect-to-x

Education

JoVE Core

Calculus

Unit

Applications of Integration

KEY TERMS AND CONCEPTS

vertical slices for area between curves

Area between curves can be found by using vertical slices and a definite integral. For two continuous functions on a closed interval from a to b, the region is bounded by their graphs on the sides and by the interval endpoints across the bottom and top edges of the setup.

A first estimate comes from splitting the interval into many narrow vertical strips of equal width. Each strip is treated like a rectangle. Its height is the vertical difference between the two functions at a chosen point inside the strip.

Adding the areas of all the rectangles gives a Riemann sum, which is a way to approximate area with many small pieces. Because the rectangle edges do not exactly follow the curves, the result is only an estimate when the number of strips is finite. As the strips become more numerous and thinner, the approximation improves.

In the limiting case, the strips become infinitely thin. Then the Riemann sum becomes a definite integral, which gives the exact area using infinitely thin vertical slices. This method is useful beyond pure math. In economics, the area between the line of perfect equality and a curve showing actual income distribution can be used to measure income inequality. The size of that area gives a numerical value for the inequality gap.

Vertical Slices for Area Between Curves

Area between curves can be found by using vertical slices and a definite integral. For two continuous functions on a closed interval from a to b, the region is bounded by their graphs on the sides and by the interval endpoints across the bottom and top edges of the setup. A first estimate comes from splitting the interval into many narrow vertical strips of equal width. Each strip is treated like a rectangle. Its height is the vertical difference between the two functions at a chosen point...

area between curves: integrating with respect to y

area-between-curves-integrating-with-respect-to-y

Horizontal Slices for Area Between Curves

Area between curves can be found with horizontal slices when both boundaries are written as functions of y. The left and right curves are continuous from y = c to y = d, and those lines set the vertical limits of the region. Because the curves depend on y, the area is best measured by moving across the shape with thin horizontal strips. The method uses a Riemann sum. The region is divided into many thin strips with the same height, delta y. For each strip, the width is the horizontal distance...

area between curves: problem solving

Piecewise Area Between Curves

Piecewise area between curves uses more than one integral to find the size of a region on a graph. In this example, the region is enclosed by a square root function, a reflected cube root function, and a linear function. The line intersects each of the other two curves, and those intersection points show where the boundary of the region changes. The area cannot be found with one setup over the full interval because the top and bottom curves are not the same everywhere. The region is split at...

calculation of volume of solids by integration

calculation-of-volume-of-solids-by-integration

Slicing Methods for Solid Volume

Slicing methods help find the volume of solids with changing cross-sections. A basic solid, like a rectangular box, is easy to measure because its base area stays the same along its height. Many real objects do not have that regular shape, so elementary geometry is not enough. The slicing method solves this by breaking a solid into many very thin slices. These slices are taken perpendicular to a chosen axis, often the x-axis. Each slice has a small thickness, written as delta x, and a...

volumes of solids of revolution

Disk Method for Rotated Region Volume

The disk method is used to find the volume of a solid formed by rotating a region around an axis. The region is first drawn in a two-dimensional plane. When that region spins around a fixed line, called the axis of revolution, it creates a three-dimensional shape with rotational symmetry. These shapes appear often in math models, physics, and engineering. When the region touches the axis of revolution, the solid has no hollow space inside. Its cross-sections are circular, so the shape can be...

finding volume using cross-sectional area

finding-volume-using-cross-sectional-area

Tetrahedron Volume from Triangle Slices

Tetrahedron volume can be found by adding the areas of triangular slices. This method uses cross-sectional area, which is the area of a flat slice taken through a 3D solid. It is useful when a solid changes shape in a regular way and a simple geometric formula is not available. A tetrahedron is a good example. It has a height h and an equilateral triangle base with side length a. The coordinate system places the origin at the top vertex, and the y-axis goes downward along the height to the...

problem solving: volume

Finding the Volume of a Jet Fuel Tank

The volume of a jet fuel tank can be found with solids of revolution. The tank is modeled by rotating a two-dimensional region around the x-axis. This creates a three-dimensional shape that is symmetric about the axis of rotation. The region runs from 0 to 2 meters, so the tank’s shape can be measured along that length. Because the boundary curve touches the axis, the disk method works well here. The disk method breaks the solid into many very thin circular slices that are perpendicular to the...

average value of a function

Estimating Function Behavior with Average Value

The average value of a function gives one number that summarizes how a quantity changes over a closed interval. It can also be viewed as the height of a rectangle whose area matches the net area under the curve. That net area includes both positive and negative parts of the function, so the result reflects the function’s overall behavior. A greenhouse temperature over a 24-hour period is a practical example of this idea. The temperature may rise and fall throughout the day, but it is still...

key terms and concepts

Learn how vertical strips and Riemann sums lead to a definite integral for finding area between curves, plus a real-world example from economics.

Vertical Slices for Area Between Curves | 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.

Vertical Slices for Area Between Curves

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