Name: hydrostatic force on dam walls with integration
Uploaded: 2026-01-12T12:24:31.000Z
Duration: 1 min 30 s
Channel: JoVE

Гидростатическое давление возникает из-за веса жидкости в покое и увеличивается пропорционально глубине.Для горизонтальной поверхности, погружённой на фиксированную глубину, давление постоянно и равно произведению плотности жидкости, ускорения, вызванного гравитацией, и глубины.Но для вертикальной поверхности, такой как поверхность плотины, каждая точка находится на разной глубине, что приводит к увеличению давления с глубиной и неравномерному распределению сил.Для расчёта общей гидростатической силы при этом изменяющемся давлении подводный участок делится на тонкие горизонтальные полосы. Каждая полоса лежит на определённой глубине и имеет площадь, равную ширине, умноженной на небольшую вертикальную толщину.Давление на каждую полосу равно плотности веса жидкости, умноженной на глубину под поверхностью, а сила на эту полосу равна давлению, умноженному на площадь.Суммирование этих сил по всем полосам приближает суммарную силу с помощью суммы Римана. Когда толщина полосы приближается к нулю, сумма становится определённым интегралом.Этот интеграл выражает общую гидростатическую силу как давление, умноженное на функцию ширины, интегрированную по затопленной глубине.Этот метод необходим в гражданском строительстве для проектирования эффективных подпорных конструкций.

Применение интегрирования для расчёта гидростатической силы

Applications of Integration to Find Hydrostatic Pressure

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applications-of-integration-to-find-hydrostatic-pressure

Education

JoVE Core

Calculus

Математический анализ

Unit

Единица

Application of Techniques of Integration

Применение методов интегрирования

KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

hydrostatic force on dam walls with integration

Hydrostatic force is the total force a fluid at rest exerts on a surface. For a horizontal surface at a fixed depth, the pressure is constant. It is found by multiplying fluid density, gravitational acceleration, and depth.

A vertical dam wall is different because pressure increases with depth. The water above each point adds to the pressure below it. That means the force is not spread evenly across the wall.

Integration helps measure this changing force. Pressure in a still fluid rises linearly with depth. At depth h, the pressure P depends on the fluid density ρ and gravity g.

To find the total force, the submerged surface is split into very thin horizontal strips. Each strip has width w(h) and small height dh, so its area is w(h)dh. The force on one strip is written as a differential force dF.

The forces on all strips are added with a definite integral from depth a to depth b. This method also works when the wall width changes with depth. It can model both flat and curved surfaces.

These calculations matter in civil and hydraulic engineering. Engineers use them to design dams, retaining walls, gates, and storage tanks safely. Integration gives the precision needed to handle changing fluid pressure and protect structural integrity.

Hydrostatic Force on Dam Walls with Integration

hydrostatic-force-on-dam-walls-with-integration

arc length of a curve

Measuring Curved Paths with Calculus

Measuring curved paths with calculus is important in engineering projects. Roller coaster tracks, cables, and railway lines often follow smooth curves instead of straight lines. Accurate path length helps with planning and design. A curve is first split into small segments. Each short segment is treated as a straight line between two nearby points on the curve. Adding these line segments gives an estimate of the total distance along the path. The estimate becomes better when the segments are...

arc length of a curve: problem solving

Catenary Cable Length in Power Lines

Catenary cable length helps engineers measure the full span of a hanging power line. A high-voltage line stretches 40 meters between two transmission towers and sags 10 meters because of gravity and thermal expansion. The cable forms a catenary, which is the curve of a uniform, flexible cable under its own weight. A catenary is not the same as a parabola. It is described by the hyperbolic cosine function, which gives a more accurate shape for a suspended cable. In this setup, engineers use the...

arc length function

Measuring Distance Along a Curve

The arc length function measures the distance traveled along a smooth curve from a fixed starting point to a moving endpoint. It is useful when a straight-line estimate is not accurate enough. For curves that are continuous and differentiable, arc length gives a precise way to track length along the curve. To find this distance, the curve is broken into many small pieces. Each piece is treated like a short straight line. Its length depends on the horizontal change and the vertical change over...

area of a surface of revolution

Calculating Surface Area from Rotation

Surfaces of revolution are created when a two-dimensional curve is rotated around an axis. The rotation produces a three-dimensional shape. In engineering, this calculation can help determine the surface area of a rocket nozzle. That information matters when a heat-resistant coating must be applied evenly. When a curve is revolved about the x-axis, it sweeps out a continuous surface. To find its area, the curve is first broken into small straight-line pieces over short intervals. Each rotated...

applications of integration to find hydrostatic pressure

Hydrostatic force is the total force a fluid at rest exerts on a surface. For a horizontal surface at a fixed depth, the pressure is constant. It is found by multiplying fluid density, gravitational acceleration, and depth. A vertical dam wall is different because pressure increases with depth. The water above each point adds to the pressure below it. That means the force is not spread evenly across the wall. Integration helps measure this changing force. Pressure in a still fluid rises...

applications of integration to find centers of mass

applications-of-integration-to-find-centers-of-mass

Center of Mass in Discrete and Continuous Systems

Center of mass can be found from rotational equilibrium and from integration. A balanced plank on a pivot shows the idea clearly. When the net torque about the pivot is zero, the system is in balance. Torque is the force on an object times its perpendicular distance from the pivot. At that point, the pivot lies at the center of mass. For a set of discrete point masses, the center of mass is the position where the masses balance by weight. Each mass contributes its mass times its position,...

theorem of pappus

Pappus Theorem for Volume and Surface Area

Pappus Theorem connects centroid motion to the volume and surface area of solids of revolution. It is also called the Pappus–Guldinus Theorem. The method uses geometry to find these values when a plane region or a plane curve is rotated about an external axis. The theorem has two related parts. The first part gives the volume of a solid formed by rotating a plane region around an axis outside the region. In this case, the volume equals the area of the region multiplied by the distance traveled...

applications of integration to find consumer surplus

applications-of-integration-to-find-consumer-surplus

Consumer Surplus from Demand and Integration

Consumer surplus measures the economic gain buyers receive when they pay less than the highest price they are willing to pay. In microeconomics, this idea comes from the demand function, which links quantity to the price consumers will pay. Because willingness to pay usually falls as quantity rises, the demand curve slopes downward. When a product is sold at a fixed price, some buyers would have paid more for the first units they bought. The difference between that willingness to pay and the...

applications of integration to find blood flow

applications-of-integration-to-find-blood-flow

Using Integration to Model Blood Flow

Blood flow in a cylindrical blood vessel can be modeled with integration and laminar flow. Laminar flow is smooth motion in parallel layers. In this model, blood speed changes with distance from the center of the vessel. The blood moves fastest along the central axis. As the distance from the center increases, the velocity decreases. At the vessel wall, the velocity drops to zero because of viscous drag. This changing speed across the cross-section is the key idea behind the calculation. To...

introduction to normal distributions

Normal Distribution in Test Scores

Normal distribution helps explain how standardized test scores can be spread across a group. It is a key idea in statistics for describing symmetry, center, and variability in large sets of data. Test score data often start as a histogram. In a histogram, each bar shows how many scores fall within a score range. To turn that count-based graph into a continuous probability model, the histogram is normalized by dividing each bar height by the total number of observations and by the bin width. <br...

applications of integration to probability density functions

applications-of-integration-to-probability-density-functions

Finding Probability with Area Under a PDF

Continuous probability distributions describe random variables that can take any real value within a set range. Height is one example, since it can be measured with growing precision, such as 163.5 or 165.25 centimeters. That makes height a continuous random variable. The pattern of a continuous random variable is shown with a probability density function, or PDF. A PDF shows how probability is spread across the possible values. In a continuous distribution, probabilities are assigned to...

key terms and concepts

Learn how integration is used to calculate hydrostatic force on dam walls by adding the changing pressure of water across submerged surfaces.

Hydrostatic Force on Dam Walls with Integration | 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

Hydrostatic Force on Dam Walls with Integration

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