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

Volumes of irregularly shaped objects can be accurately determined using the concept of solids of revolution. For example, consider a curved function in a two-dimensional plane. Rotating this curve about the x-axis generates a symmetrical three-dimensional shape known as a solid of revolution. The line around which the region revolves is the axis of revolution.When the region lies directly against the axis, it produces solid circular cross-sections. To determine the volume of such a solid, the disk method may be used.This method divides the solid into thin slices perpendicular to the axis. Each slice becomes a disk with a small width and radius equal to the distance from the axis to the curve.The volume of each disk is obtained by multiplying its area by its thickness. The total volume of the solid is approximated by summing the volumes of all the circular disks formed.As the number of disks increases and their thickness approaches zero, the summation transitions into the definite integral of the cross-sectional area over the given interval, yielding the exact volume.The variable of integration and the expression for radius depend on the orientation of the axis.

volumes of solids of revolution

Volumes of Solids of Revolution

Volumes of irregularly shaped objects can be systematically determined using the concept of solids of revolution. This approach begins with a region defined by a curve in a two-dimensional plane. When this region is rotated about a fixed line, known as the axis of revolution, it generates a three-dimensional object with rotational symmetry. Such objects frequently arise in mathematical modeling, physics, and engineering applications.When the region being rotated lies directly against the axis of revolution, the resulting solid consists of circular cross-sections with no hollow interior. In this case, the disk method provides an effective technique for computing the volume. The solid is conceptually divided into a large number of very thin slices taken perpendicular to the axis of rotation. Each slice has a small thickness and forms a disk whose radius is determined by the distance from the axis to the boundary curve at that location.Each disk represents a simple geometric solid whose volume can be approximated by multiplying its circular area by its thickness. Although an individual disk only approximates a portion of the solid, the collection of disks closely represents the entire object. Adding the volumes of all disks produces an approximation of the total volume. As the slices become increasingly thin, this approximation improves in accuracy.In the limiting case, where the thickness of each disk approaches zero and the number of slices becomes very large, the summation of disk volumes becomes an exact calculation through definite integration. The integral accumulates the cross-sectional areas along the length of the solid, yielding the precise volume of the solid of revolution. The choice of variable of integration, as well as the expression used to represent the disk radius, depends on whether the rotation occurs about a horizontal or vertical axis.The disk method is a foundational tool in calculus and is widely applied to problems involving smooth, solid shapes generated by rotation. Its conceptual simplicity and geometric interpretation make it particularly valuable for analyzing physical systems and engineered components with rotational symmetry.

Volumes of irregularly shaped objects can be systematically determined using the concept of solids of revolution. This approach begins with a region ...

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Calculus

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Applications of Integration

KEY TERMS AND CONCEPTS

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

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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...

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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.

Disk Method for Rotated Region Volume

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