Name: volume of curved tanks with multiple integrals
Uploaded: 2026-06-08T09:39:07.383Z
Duration: 1 min 18 s
Channel: JoVE

Consider a paraboloid-shaped water tank filled with water. The water level has a depth of 4 meters and a diameter of 4 meters at the top.The objective is to calculate the total volume of water using a triple integral in cylindrical coordinates. Here, the paraboloid region S can be modeled by the coordinates r, theta and z.The integration process starts vertically along the z-axis.Here the volume is bounded above by the plane z equals 4 and below by the paraboloid surface z equals r squared.As r goes between its lower limit of zero and upper limit of 2, the former integral with respect to z accounts for the changing height.Together, as theta remains fixed, r and z then sweep out a flat two-dimensional vertical slice.Finally, this two-dimensional cross-section is rotated through a complete circle of 2 pi radians.This rotation sweeps the slice through every angular position, encompassing the entire three-dimensional volume of the tank.

real-life applications of multiple integrals

Real-Life Applications of Multiple Integrals

Multiple integrals provide a powerful mathematical framework for calculating physical quantities distributed throughout two- and three-dimensional regions. One important application is the determination of volume in objects with curved geometries, such as storage tanks, pipes, and reservoirs. Cylindrical coordinates are especially useful for systems with rotational symmetry because they simplify the description of circular and paraboloid-shaped regions.Consider a paraboloid-shaped water tank filled with water to a depth of 4 meters and having a top diameter of 4 meters. Because the tank is symmetric about the vertical axis, cylindrical coordinates provide a natural way to describe the region. In this coordinate system, the position of each point is determined by the radial distance r, the angular coordinate &#x3B8;, and the vertical coordinate z.The integration process begins along the vertical direction. At a fixed radial position, the height extends upward from the curved paraboloid surface to the water level at the top of the tank. As the radial distance increases from the center outward to the boundary of the tank, the vertical height changes continuously. This variation in height is incorporated directly into the integration process, allowing the curved geometry of the tank to be accurately represented.For a fixed angular position, the radial and vertical coordinates define a two-dimensional slice of the tank. Rotating this slice through all angular positions around the central axis generates the complete three-dimensional region occupied by the water. The triple integral accumulates the contribution from every infinitesimal portion of the tank, producing the total volume. This approach demonstrates how multiple integrals can model realistic engineering structures with nonuniform curved boundaries.

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Education

Lehre

JoVE Core

Calculus

Analysis

Unit

Einheit

Multiple Integrals and Applications

Mehrfachintegrale und Anwendungen

KEY TERMS AND CONCEPTS

KERNBEGRIFFE UND KONZEPTE

Praxisnahe Anwendungen mehrerer Integrale

volume of curved tanks with multiple integrals

Multiple integrals can be used to find the volume of curved three-dimensional objects. This method is useful for shapes such as storage tanks, pipes, and reservoirs. It works well when the region changes shape across space.

Cylindrical coordinates are especially helpful for objects with rotational symmetry. In this system, each point is described by the radial distance r, the angle θ, and the vertical coordinate z. These coordinates make circular and paraboloid-shaped regions easier to describe.

A paraboloid-shaped water tank gives a clear example. The tank is filled with water to a depth of 4 meters and has a top diameter of 4 meters. Because the tank is symmetric about the vertical axis, cylindrical coordinates fit the shape naturally.

The integration starts in the vertical direction. At a fixed radial position, the water height runs from the curved paraboloid surface up to the water level at the top of the tank. As r moves from the center to the edge, the height changes continuously, and that change is built into the integral.

For one fixed angle, the radial and vertical coordinates form a two-dimensional slice of the tank. Rotating that slice through all angles around the central axis creates the full three-dimensional region. The triple integral adds up every tiny part of the tank and gives the total volume. This shows how multiple integrals can model realistic engineering shapes with curved boundaries.

Volume of Curved Tanks with Multiple Integrals

volume-of-curved-tanks-with-multiple-integrals

introduction to double integrals

Double Integrals for Volume and Density

Double integrals help find the volume under a surface and over a region in the xy-plane. They extend single-variable integration, which finds area under a curve on an interval, to functions of two variables. This makes them useful for models in mathematics, physics, and engineering that need three-dimensional thinking. Single-variable integration uses thin rectangles to approximate area under y = f(x) from a to b. The definite integral is written as a limit of a sum of these rectangle areas.

the midpoint rule for double integrals

Estimating Surface Volume with Midpoints

The midpoint rule estimates volume for a double integral over a rectangular region. It is useful when a surface has a continuously changing height. In civil engineering, it can help estimate how much soil must be moved when planning a road across uneven terrain. The road footprint is treated as a rectangle in the xy-plane. The terrain above a flat reference level is described by a continuous height function, f(x, y). The goal is to estimate the volume of soil between that surface and the...

iterated integrals and fubini's theorem

Double Integrals as Cross-Sections

Double integrals help find the volume under a surface z = f(x,y) over a region in the plane. They extend the idea of a single-variable integral to functions of two variables. For a rectangular region with a ≤ x ≤ b and c ≤ y ≤ d, and for functions that are continuous on that region, the double integral can be written as an iterated integral. This breaks one harder problem into two simpler steps. A useful way to understand this method is through cross-sectional area. If x is held constant, then...

average value on a rectangle

Estimating Snowfall with Midpoint Rule

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double integrals over general regions

Double Integrals on Curved Boundaries

Double integrals can measure quantities spread across a two-dimensional region, such as rainfall over a lake, heat on a metal plate, or population density over land. In many real situations, the region does not have straight sides. It may have curved or irregular edges instead. That makes the region harder to describe as a simple rectangle. To handle an irregular region D, it is placed inside a larger rectangular region R. A function f(x,y) gives the quantity on D, such as rainfall intensity...

changing the order of integration in double integrals

changing-the-order-of-integration-in-double-integrals

Finding Volume with Double Integral Bounds

Double integrals can be used to find the volume of liquid in tanks with irregular sides. The base of the tank is divided into many very small rectangular sections. Each section becomes the base of a thin column of liquid, and the volumes of all the columns are added across the region. This gives an accurate estimate of the total liquid volume inside the tank. The limits of integration come from the boundaries of the tank’s base. These limits depend on how the region is divided during the...

double integrals in polar coordinates

Polar Area Elements for Double Integrals

Double integrals use polar coordinates to measure area and other quantities over two-dimensional regions with circular symmetry. This approach is useful for curved shapes that appear in engineering and design, such as ponds, reservoirs, and circular foundations. Polar coordinates describe each point by its distance from the center and its angle, which often makes the region easier to set up for integration. A semicircular pond with radius L shows why polar coordinates can be simpler than...

surface area calculations

Calculating Curved Dish Surface Area

Calculating curved dish surface area is an important step in engineering for shapes like satellite dishes. A parabolic dish can reflect communication signals efficiently, but engineers still need its exact curved surface area. That value helps them estimate coating material, fabrication costs, and structural requirements. Because the rim of the dish makes a circular boundary, the calculation is set up over a circular domain in the xy-plane. A surface given by z = f(x, y) can be written in...

triple integrals in rectangular coordinates

triple-integrals-in-rectangular-coordinates

Triple Integrals for Volume and Mass

Triple integrals help find the total value of a function across a three-dimensional region. They are often used to calculate volume, mass, and other physical quantities that change from place to place. The idea is to split a solid region into many small rectangular boxes, evaluate the function in each box, and add the results. As the boxes get smaller, the triple Riemann sum gets closer to the exact triple integral. In rectangular coordinates, the integral is written as an iterated integral,...

triple integrals over general regions

Evaluating Triple Integrals by Region

Triple integrals over bounded solid regions are used to measure a continuous function across a three-dimensional shape. The solid region E is often placed inside a rectangular box B. To handle the region cleanly, the function f(x,y,z) is extended to a new function F that matches f inside E and is zero outside the solid. The triple integral is written as ∭_E f(x,y,z) dV. For the integral to exist, the function f must be continuous. The boundary of E also needs to be smooth enough. As with...

changing the order of integration in triple integrals

changing-the-order-of-integration-in-triple-integrals

Setting Triple Integral Bounds for Easier Evaluation

Triple integrals can be easier to evaluate when the bounds are chosen to match the shape of the solid. In this example, the region is enclosed by a flat base, a slanted plane, two vertical planes, and a parabolic cylinder. The task is to integrate e x over this three-dimensional region, so the goal is to describe the limits in the simplest useful order. One setup uses x as the innermost variable. In that case, each line segment inside the solid runs in the x-direction. The lower limit for x...

triple integrals in cylindrical coordinates

triple-integrals-in-cylindrical-coordinates

Volume of Cylinders with Triple Integrals

Cylindrical coordinates describe a point in three-dimensional space with three values: radial distance, angle, and height. The height gives the position above the xy-plane. The radial distance tells how far the point is from the z-axis, and the angle gives its direction from the positive x-axis in the xy-plane. This system is useful for regions with circular symmetry. It matches the shape of cylinders, disks, and circular tanks. For that reason, it is a natural choice when finding volume in...

triple integrals in spherical coordinates

triple-integrals-in-spherical-coordinates

Finding Sphere Volume with Spherical Wedges

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change of variables in multiple integrals

change-of-variables-in-multiple-integrals

Using the Jacobian to Simplify Integrals

Multiple integrals are used to find areas, volumes, mass distributions, and other physical quantities over regions in two or three dimensions. In many problems, the original region has curved boundaries when written in Cartesian coordinates. Those boundaries can make the limits of integration hard to describe and the calculation more difficult. A change of variables helps simplify the setup. It replaces the original coordinates x and y with new variables u and v. The original region R in the...

moments and centers of mass

Balance Point of a Variable-Density Plate

A flat metal plate, or lamina, balances at a point called its center of mass. That balance point depends on the plate’s shape and on how mass is spread through the material. If the density changes from place to place, some regions affect the balance more than others. The center of mass can be treated as a weighted average of every point in the plate. For a plate in the xy-plane, the density at each point is written as ρ(x, y). To study the mass distribution, the region D is divided into many...

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Changing variables in double integrals can make a hard region much easier to integrate over. This method is useful when finding physical quantities spread across a two-dimensional area, such as the total mass of an elliptical plate. The density function is evaluated at every point in the region, and all of those contributions are added together to get the mass. If the integral is set up in rectangular coordinates, an ellipse often creates difficult limits. Those limits can involve square root...

Multiple integrals can be used to find the volume of curved three-dimensional objects. This method is useful for shapes such as storage tanks, pipes, and reservoirs. It works well when the region changes shape across space. Cylindrical coordinates are especially helpful for objects with rotational symmetry. In this system, each point is described by the radial distance r, the angle θ, and the vertical coordinate z. These coordinates make circular and paraboloid-shaped regions easier to...

key terms and concepts

Learn how multiple integrals and cylindrical coordinates help find the volume of curved tanks, including a paraboloid-shaped water tank with rotational symmetry.

Volume of Curved Tanks with Multiple Integrals | 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

Volume of Curved Tanks with Multiple Integrals

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