Name: slicing methods for solid volume
Uploaded: 2026-01-12T12:24:41.000Z
Duration: 1 min 27 s
Channel: JoVE

Le calcul du volume commence souvent avec des solides simples. Pour une boîte rectangulaire, le volume est déterminé en multipliant la surface de sa base, donnée par longueur multipliée par largeur, par sa hauteur.Cependant, de nombreux solides complexes n’ont pas une section transversale uniforme. Leur volume doit être calculé à l’aide d’une technique puissante appelée la méthode de découpage, qui repose sur l’intégration.Dans la méthode de tranchement, le solide est traité comme une pile de tranches très fines prises perpendiculairement à un axe choisi.Chaque tranche est approximée comme un solide mince dont l’aire de section transversale est donnée par une fonction de la position de la tranche, x.Le volume de chaque tranche est approximativement A(x) multiplié par Δx. La somme de ces approximations donne une somme de Riemann, qui devient une intégrale définie à mesure que les tranches s’affinent.En conséquence, le volume total est calculé en intégrant la fonction de section transversale entre les points de départ et de fin du solide.Cette méthode est utilisée en ingénierie pour calculer les volumes de structures irrégulières, telles que les barrages en béton, en intégrant les surfaces de section transversale sur toute la longueur. Le volume calculé aide à estimer les matériaux de construction nécessaires.

calcul du volume des solides par intégration

Calculation of Volume of Solids by Integration

Le calcul du volume commence souvent par l&#x2019;&#xE9;tude de solides g&#xE9;om&#xE9;triques simples. Par exemple, le volume d&#x2019;un parall&#xE9;l&#xE9;pip&#xE8;de rectangle s&#x2019;obtient en multipliant l&#x2019;aire de sa base par sa hauteur. Cette approche directe repose sur le fait que l&#x2019;aire de la section transversale du parall&#xE9;l&#xE9;pip&#xE8;de reste constante sur toute sa longueur. Toutefois, de nombreux objets du monde r&#xE9;el ne pr&#xE9;sentent pas de sections transversales uniformes, et leur volume ne peut donc pas &#xEA;tre d&#xE9;termin&#xE9; &#xE0; l&#x2019;aide de formules g&#xE9;om&#xE9;triques &#xE9;l&#xE9;mentaires.
Pour rem&#xE9;dier &#xE0; cette limitation, la m&#xE9;thode des tranches fournit une approche syst&#xE9;matique permettant de calculer le volume de solides complexes &#xE0; l&#x2019;aide de l&#x2019;int&#xE9;gration. Dans cette approche, le solide est con&#xE7;u comme &#xE9;tant constitu&#xE9; d&#x2019;un grand nombre de tranches tr&#xE8;s fines, prises perpendiculairement &#xE0; un axe choisi, g&#xE9;n&#xE9;ralement l&#x2019;axe des abscisses. Chaque tranche est suppos&#xE9;e avoir une faible &#xE9;paisseur, not&#xE9;e &#x394;x, et une aire de section transversale qui d&#xE9;pend de sa position le long de l&#x2019;axe. Bien que chaque tranche ne constitue qu&#x2019;une approximation de la forme r&#xE9;elle, elle ressemble &#xE9;troitment &#xE0; un solide simple dont le volume peut &#xEA;tre estim&#xE9;.
Le volume d&#x2019;une tranche individuelle est approxim&#xE9; en multipliant son aire de section transversale par son &#xE9;paisseur. Lorsque les volumes de toutes les tranches sont additionn&#xE9;s, on obtient une somme de Riemann qui approxime le volume total du solide. &#xC0; mesure que les tranches deviennent plus fines et plus nombreuses, cette approximation s&#x2019;am&#xE9;liore. &#xC0; la limite, la somme de Riemann converge vers une int&#xE9;grale d&#xE9;finie, ce qui permet d&#x2019;obtenir le volume exact du solide en accumulant les aires des sections transversales sur l&#x2019;ensemble de l&#x2019;intervalle, du d&#xE9;but &#xE0; l&#x2019;extr&#xE9;mit&#xE9; de l&#x2019;objet.
La m&#xE9;thode des tranches est largement utilis&#xE9;e en g&#xE9;nie et en sciences appliqu&#xE9;es. Un exemple courant est le calcul du volume de structures irr&#xE9;guli&#xE8;res telles que les barrages en b&#xE9;ton. La forme de la section transversale d&#x2019;un barrage varie souvent le long de sa longueur en raison de contraintes structurelles et g&#xE9;ologiques. En mod&#xE9;lisant l&#x2019;aire de la section transversale comme une fonction de la position et en l&#x2019;int&#xE9;grant sur toute la longueur du barrage, les ing&#xE9;nieurs peuvent estimer avec pr&#xE9;cision le volume total. Cette information est essentielle pour d&#xE9;terminer les besoins en mat&#xE9;riaux, les co&#xFB;ts de construction et la faisabilit&#xE9; globale du projet.

Calcul du volume des solides par intégration

Le calcul du volume commence souvent par l&#x2019;&#xE9;tude de solides g&#xE9;om&#xE9;triques simples. Par exemple, le volume d&#x2019;un ...

calculation-of-volume-of-solids-by-integration

Education

Enseignement

JoVE Core

Calculus

Calcul

Unit

Unité

Applications of Integration

Applications de l'intégration

KEY TERMS AND CONCEPTS

TERMES ET CONCEPTS CLÉS

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 cross-sectional area that changes with position.

Each thin slice is treated like a simple shape whose volume can be estimated. The volume of one slice is found by multiplying its cross-sectional area by its thickness. When all of the slice volumes are added, the result is a Riemann sum, which gives an approximation of the whole solid.

As the slices become thinner and more numerous, the approximation becomes more accurate. In the limit, the Riemann sum becomes a definite integral. That integral gives the exact volume by adding the cross-sectional areas across the full interval of the object.

Engineers and applied scientists use this method for irregular structures such as concrete dams. A dam may change shape along its length because of structural and geological needs. By modeling cross-sectional area as a function of position and integrating along the dam, engineers can estimate total volume and use that result to plan material needs, construction costs, and project feasibility.

Slicing Methods for Solid 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

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 the slicing method uses cross-sectional area, Riemann sums, and definite integrals to estimate the volume of irregular solids like concrete dams.

Slicing Methods for Solid 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

Slicing Methods for Solid Volume

Description

Transcript

Tags