Name: disk method for rotated region volume
Uploaded: 2026-01-12T12:24:09.000Z
Duration: 1 min 29 s
Channel: JoVE

不规则形状物体的体积可以通过旋转体的概念准确测定。例如，考虑二维平面上的一条曲线函数。将该曲线绕x轴旋转，可生成一个称为旋转体的对称三维形状。区域绕其旋转的直线称为旋转轴。当区域紧邻轴时，会形成实心的圆形横截面。为了确定此类立体的体积，可使用圆盘法。该方法将立体沿垂直于轴的方向切割成若干薄片。每个薄片成为一个圆盘，其宽度很小，半径等于从轴到曲线的距离。通过将每个圆盘的面积乘以其厚度，可得到每个圆盘的体积。通过将所有形成的圆形圆盘的体积相加，可近似得到该立体的总体积。随着圆盘数量的增加且其厚度趋近于零，求和过程转化为在给定区间上对横截面积的定积分，从而得到精确的体积。积分变量和半径的表达式取决于轴的方向。

旋转体的体积

Volumes of Solids of Revolution

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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

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 broken into many thin disks. Each disk is taken perpendicular to the axis of rotation. The radius of each disk comes from the distance between the axis and the curve at that point.

Each disk has a small thickness and a circular area. Its volume can be estimated by multiplying that area by the thickness. One disk gives only an approximation, but many disks together give a close model of the whole solid. As the slices become thinner and more numerous, the estimate becomes more accurate.

In the limit, this sum of disk volumes becomes an exact volume through definite integration. The integral adds the cross-sectional areas along the length of the solid. The variable of integration and the radius expression depend on whether the axis of rotation is horizontal or vertical. This method is a key idea in calculus for finding the volume of smooth, rotated shapes.

Disk Method for Rotated Region Volume

area between curves: integrating with respect to x

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Vertical Slices for Area Between Curves

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area between curves: integrating with respect to y

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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...

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Piecewise Area Between Curves

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calculation of volume of solids by integration

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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

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

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Tetrahedron Volume from Triangle Slices

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problem solving: volume

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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 the disk method finds the volume of a solid of revolution by slicing a rotated region into thin disks and using definite integration.

Disk Method for Rotated Region Volume | 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

Disk Method for Rotated Region Volume

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