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

Les volumes d’objets de forme irrégulière peuvent être déterminés avec précision en utilisant le concept de solides de révolution. Par exemple, considérons une fonction courbe dans un plan bidimensionnel. Faire tourner cette courbe autour de l’axe des x donne une forme symétrique tridimensionnelle appelée solide de révolution. La ligne autour de laquelle la région tourne est l’axe de la révolution.Lorsque la région est directement opposée à l’axe, elle produit des sections transversales circulaires solides. Pour déterminer le volume d’un tel solide, la méthode du disque peut être utilisée.Cette méthode divise le solide en fines tranches perpendiculaires à l’axe. Chaque tranche devient un disque d’une petite largeur et d’un rayon égaux à la distance entre l’axe et la courbe.Le volume de chaque disque est obtenu en multipliant sa surface par son épaisseur. Le volume total du solide est approximé en additionnant les volumes de tous les disques circulaires formés.Lorsque le nombre de disques augmente et que leur épaisseur tend vers zéro, la somme se transforme en l’intégrale définie de l’aire de la section transversale sur l’intervalle donné, donnant le volume exact.La variable d’intégration et l’expression pour le rayon dépendent de l’orientation de l’axe.

volumes des solides de révolution

Volumes of Solids of Revolution

Le volume d&#x2019;objets de forme irr&#xE9;guli&#xE8;re peut &#xEA;tre d&#xE9;termin&#xE9; de mani&#xE8;re syst&#xE9;matique &#xE0; l&#x2019;aide du concept de solides de r&#xE9;volution. Cette approche commence par une r&#xE9;gion d&#xE9;limit&#xE9;e par une courbe dans un plan bidimensionnel. Lorsque cette r&#xE9;gion est tourn&#xE9;e autour d&#x2019;une droite fixe, appel&#xE9;e axe de r&#xE9;volution, elle engendre un objet tridimensionnel pr&#xE9;sentant une sym&#xE9;trie de rotation. De tels objets apparaissent fr&#xE9;quemment en mod&#xE9;lisation math&#xE9;matique, en physique et dans des applications d&#x2019;ing&#xE9;nierie.
Lorsque la r&#xE9;gion mise en rotation est au contact de l&#x2019;axe de r&#xE9;volution, le solide obtenu est constitu&#xE9; de sections circulaires pleines, sans cavit&#xE9; int&#xE9;rieure. Dans ce cas, la m&#xE9;thode des disques constitue une technique efficace pour calculer le volume. Le solide est divis&#xE9; conceptuellement en un tr&#xE8;s grand nombre de tranches extr&#xEA;mement fines, prises perpendiculairement &#xE0; l&#x2019;axe de rotation. Chaque tranche, de faible &#xE9;paisseur, forme un disque dont le rayon est d&#xE9;termin&#xE9; par la distance entre l&#x2019;axe et la courbe de d&#xE9;limitation en ce point.
Chaque disque repr&#xE9;sente un solide g&#xE9;om&#xE9;trique simple dont le volume peut &#xEA;tre approxim&#xE9; en multipliant l&#x2019;aire du disque par son &#xE9;paisseur. Bien qu&#x2019;un disque pris isol&#xE9;ment ne fournisse qu&#x2019;une approximation d&#x2019;une portion du solide, l&#x2019;ensemble des disques repr&#xE9;sente de pr&#xE8;s l&#x2019;objet dans sa totalit&#xE9;. L&#x2019;addition des volumes de tous les disques fournit une approximation du volume total. &#xC0; mesure que les tranches deviennent de plus en plus fines, cette approximation gagne en pr&#xE9;cision.
Dans le cas limite, lorsque l&#x2019;&#xE9;paisseur de chaque disque tend vers z&#xE9;ro et que le nombre de tranches devient tr&#xE8;s grand, la somme des volumes des disques devient un calcul exact par int&#xE9;gration d&#xE9;finie. L&#x2019;int&#xE9;grale additionne les aires des sections transversales le long de la longueur du solide, fournissant ainsi le volume exact du solide de r&#xE9;volution. Le choix de la variable d&#x2019;int&#xE9;gration, ainsi que l&#x2019;expression utilis&#xE9;e pour repr&#xE9;senter le rayon du disque, d&#xE9;pend du fait que la rotation s&#x2019;effectue autour d&#x2019;un axe horizontal ou vertical.
La m&#xE9;thode des disques constitue un outil fondamental du calcul diff&#xE9;rentiel et int&#xE9;gral et se trouve largement utilis&#xE9;e pour l&#x2019;&#xE9;tude de probl&#xE8;mes portant sur des formes solides lisses engendr&#xE9;es par rotation. Sa simplicit&#xE9; conceptuelle et son interpr&#xE9;tation g&#xE9;om&#xE9;trique en font une m&#xE9;thode particuli&#xE8;rement pr&#xE9;cieuse pour l&#x2019;analyse de syst&#xE8;mes physiques et de composants techniques pr&#xE9;sentant une sym&#xE9;trie de rotation.

Volumes des solides de révolution

Le volume d&#x2019;objets de forme irr&#xE9;guli&#xE8;re peut &#xEA;tre d&#xE9;termin&#xE9; de mani&#xE8;re syst&#xE9;matique &#xE0; l&#x2019;aide du ...

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Applications of Integration

Applications de l'intégration

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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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