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

Volumes de objetos de formato irregular podem ser determinados com precisão usando o conceito de sólidos de revolução. Por exemplo, considere uma função curva em um plano bidimensional. Girar essa curva em torno do eixo x gera uma forma tridimensional simétrica conhecida como sólido de revolución. A linha em torno da qual a região gira é o eixo da revolução.Quando a região está diretamente contra o eixo, ela produz seções transversais circulares sólidas. Para determinar o volume de tal sólido, pode-se usar o método do disco.Esse método divide o sólido em fatias finas perpendiculares ao eixo. Cada fatia se torna um disco com uma largura e raio pequenos iguais à distância do eixo até a curva.O volume de cada disco é obtido multiplicando sua área por sua espessura. O volume total do sólido é aproximado somando os volumes de todos os discos circulares formados.À medida que o número de discos aumenta e sua espessura se aproxima de zero, a soma transita para a integral definida da área da seção transversal ao longo do intervalo dado, resultando no volume exato.A variável de integração e a expressão para o raio dependem da orientação do eixo.

volumes de sólidos de revolução

Volumes of Solids of Revolution

Os volumes de objetos com formas irregulares podem ser determinados sistematicamente por meio do conceito de s&#xF3;lidos de revolu&#xE7;&#xE3;o. Essa abordagem come&#xE7;a com uma regi&#xE3;o definida por uma curva em um plano bidimensional. Quando essa regi&#xE3;o &#xE9; rotacionada em torno de uma linha fixa, conhecida como eixo de revolu&#xE7;&#xE3;o, ela gera um objeto tridimensional com simetria rotacional. Tais objetos surgem frequentemente em modelagem matem&#xE1;tica, f&#xED;sica e aplica&#xE7;&#xF5;es de engenharia.
Quando a regi&#xE3;o a ser rotacionada est&#xE1; em contato direto com o eixo de revolu&#xE7;&#xE3;o, o s&#xF3;lido resultante consiste em se&#xE7;&#xF5;es transversais circulares sem cavidade interna. Nesse caso, o m&#xE9;todo dos discos fornece uma t&#xE9;cnica eficaz para calcular o volume. O s&#xF3;lido &#xE9; conceitualmente dividido em um grande n&#xFA;mero de fatias muito finas, perpendicular ao eixo de rota&#xE7;&#xE3;o. Cada fatia tem uma pequena espessura e forma um disco cujo raio &#xE9; determinado pela dist&#xE2;ncia do eixo &#xE0; curva limite naquele ponto.
Cada disco representa um s&#xF3;lido geom&#xE9;trico simples cujo volume pode ser aproximado pela multiplica&#xE7;&#xE3;o de sua &#xE1;rea circular por sua espessura. Embora um disco individual aproxime apenas uma por&#xE7;&#xE3;o do s&#xF3;lido, o conjunto de discos representa de forma bastante pr&#xF3;xima o objeto inteiro. A soma dos volumes de todos os discos produz uma aproxima&#xE7;&#xE3;o do volume total. &#xC0; medida que as fatias se tornam cada vez mais finas, essa aproxima&#xE7;&#xE3;o melhora em precis&#xE3;o.
No caso limite, em que a espessura de cada disco se aproxima de zero e o n&#xFA;mero de fatias se torna muito grande, a soma dos volumes dos discos se torna um c&#xE1;lculo exato por meio de integra&#xE7;&#xE3;o definida. A integral acumula as &#xE1;reas das se&#xE7;&#xF5;es transversais ao longo do comprimento do s&#xF3;lido, resultando no volume preciso do s&#xF3;lido de revolu&#xE7;&#xE3;o. A escolha da vari&#xE1;vel de integra&#xE7;&#xE3;o, bem como a express&#xE3;o usada para representar o raio do disco, depende de a rota&#xE7;&#xE3;o ocorrer em torno de um eixo horizontal ou vertical.
O m&#xE9;todo dos discos &#xE9; uma ferramenta fundamental no C&#xE1;lculo e &#xE9; amplamente aplicado a problemas que envolvem formas s&#xF3;lidas suaves geradas por rota&#xE7;&#xE3;o. Sua simplicidade conceitual e interpreta&#xE7;&#xE3;o geom&#xE9;trica o tornam particularmente valioso para a an&#xE1;lise de sistemas f&#xED;sicos e componentes de engenharia com simetria rotacional.

Volumes de sólidos de revolução

Os volumes de objetos com formas irregulares podem ser determinados sistematicamente por meio do conceito de s&#xF3;lidos de revolu&#xE7;&#xE3;o. Essa ...

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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

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

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

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 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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