Name: finding volume with double integral bounds
Uploaded: 2026-06-08T09:38:38.922Z
Duration: 1 min 21 s
Channel: JoVE

Double integrals can be used to calculate the total volume of liquid contained in a tank with irregularly shaped sides.To determine this volume mathematically, the base region of the tank is divided into many small rectangular columns, and the volumes of these columns are summed using a double integral.The sides of the tank can be used to find the limits of integration. These limits depend on how the base region is sliced.One method is to describe the base using strips parallel to the x-axis. In this approach, integration is performed first along the x-axis and then along the y-axis.But when complex curves define the boundaries of the tank, the limits for these strips may become difficult to express.To simplify the calculation, the same region can be described using strips parallel to the y-axis. This reverses the order of integration.After rewriting the limits in the new order, the integral still represents the same physical region of the tank.Although the order changes, the region remains the same, and the new limits make the calculation easier.

changing the order of integration in double integrals

Double integrals provide a practical method for determining the volume of liquid contained in tanks with irregularly shaped sides. To estimate the total volume, the base region of the tank is divided into many very small rectangular sections. Each section forms the base of a thin column of liquid, and the volumes of all these columns are added together across the entire region. This process produces an accurate representation of the total liquid volume inside the tank.The limits of integration are determined from the boundaries of the tank&#x2019;s base. These limits depend on the way the region is divided during the calculation. One common approach is to describe the region using strips parallel to the x-axis. In this method, integration is carried out first across the horizontal direction and then across the vertical direction. For regions bounded by simple curves, this setup is often straightforward.However, when the tank boundaries are formed by more complicated curves, expressing the limits for these strips may become difficult. In such situations, the same region can instead be described using strips parallel to the y-axis. This reverses the order of integration and often leads to simpler boundary descriptions.Although the order of integration changes, the physical region represented by the integral remains exactly the same. The total volume obtained is therefore unchanged. Reversing the order simply provides a more convenient mathematical description of the region, making the evaluation of the double integral easier and more efficient for irregularly shaped tanks.

Changing the Order of Integration in Double Integrals

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Multiple Integrals and Applications

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Cambiamento dell'ordine di integrazione negli integrali doppi

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 calculation. One common method uses strips parallel to the x-axis. In that setup, the integration goes first across the horizontal direction and then across the vertical direction. For regions bounded by simple curves, this arrangement is often easy to write.

When the tank boundaries are formed by more complicated curves, the limits for x-axis strips can be hard to describe. In that case, the same region can be written using strips parallel to the y-axis instead. This changes the order of integration and often makes the boundary equations simpler. The physical region stays the same, so the total volume does not change. Reversing the order only gives a more convenient way to describe the region and evaluate the double integral.

Finding Volume with Double Integral Bounds

finding-volume-with-double-integral-bounds

introduction to double integrals

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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...

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key terms and concepts

Learn how double integrals estimate liquid volume in irregular tanks and how switching between x-axis and y-axis strips can make the bounds easier to set.

Finding Volume with Double Integral Bounds | 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

Finding Volume with Double Integral Bounds

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