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

考虑闭区间 [a,b] 上的两个连续函数。区域 𝑆 位于两条曲线之间，其上下边界由这两个函数确定，左右边界由 a 和 b 确定。可以通过将该区域划分为 n 个等宽的垂直条带来估算曲线之间的面积。每个条带被视为一个矩形，这在黎曼和方法中使用。考虑第 i条 矩形条。取其底边，以及在 xi 处的高度。该矩形的面积通过将其高度乘以底边计算得到。该区域的总面积估计值可通过将所有 n 个矩形的面积相加得到。这些矩形并不能完全精确地覆盖整个区域。但当 n 趋近于无穷大且矩形数量增加时，求和中的总面积趋近于精确值。此极限被定义为积分，并给出曲线之间区域的精确面积。这一概念在经济学中用于计算不平等差距，即完全平等线与实际收入分配曲线之间区域的面积。

曲线间面积：关于 x 的积分

Area Between Curves: Integrating With Respect to x

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&#x8003;&#x8651;&#x5728;&#x95ED;&#x533A;&#x95F4; [a, b] ...

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

Education

教育

JoVE Core

Calculus

微积分

Unit

单元

Applications of Integration

积分的应用

KEY TERMS AND CONCEPTS

关键术语与概念

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: integrating with respect to x

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.

calculus

Vertical Slices for Area Between Curves

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