Name: finding the volume of a jet fuel tank
Uploaded: 2026-01-12T12:24:49.000Z
Duration: 1 min 13 s
Channel: JoVE

A fuel tank mounted on the wing of a jet aircraft is formed by rotating a region about the central axis. This region is formed by rotating a mathematical function about the x-axis and extends from zero to two meters.To find the volume of the tank, the disk method is used, which involves slicing the solid into infinitesimally thin circular disks perpendicular to the x-axis.Each disk has an area equal to &#55349;&#57099; times the square of the function's value. The total volume is found by integrating these areas over the interval.After squaring the function, the integrand simplifies to a constant multiplied by the second power of x and the difference between two and x.Expanding and integrating this expression produces an antiderivative involving the third and fourth powers of x.Evaluating the definite integral from zero to two and substituting the limits gives an expression. Further simplifying yields a volume of approximately 1 cubic meter, which is the total volume of the fuel tank.

problem solving: volume

Problem Solving: Volume

The volume of a fuel tank mounted on the wing of a jet aircraft can be modeled using the concept of solids of revolution. In this case, the tank is formed by rotating a two-dimensional region, defined by a mathematical function, about the x-axis. The region extends along the axis from zero to two meters, and the resulting three-dimensional shape is symmetric about the axis of rotation. Because the boundary curve lies directly against the axis, the disk method is an appropriate technique for determining the volume.Using the disk method, the solid is conceptually divided into an infinite number of extremely thin circular slices taken perpendicular to the x-axis. Each slice forms a disk whose radius is equal to the value of the defining function at that position. The area of each disk is proportional to &#960; multiplied by the square of the radius. Although each individual disk represents only a small portion of the tank, the collection of all disks closely approximates the entire volume.To determine the total volume, the areas of all disks are accumulated along the length of the tank by integration. After squaring the function that defines the tank&#8217;s shape, the resulting expression simplifies to a constant multiplied by the square of the horizontal position and the difference between two and that position. This expression is then expanded, producing terms involving the third and fourth powers of the variable. Integrating these terms yields an antiderivative that describes how volume accumulates along the axis of integration.Evaluating the definite integral between zero and two meters and substituting the limits produces a numerical result. After simplification, the calculated volume is approximately one cubic meter. This value represents the total internal capacity of the fuel tank. Such calculations are crucial in aerospace engineering, where precise volume estimates are necessary to determine fuel capacity, weight distribution, and overall aircraft performance.

The volume of a fuel tank mounted on the wing of a jet aircraft can be modeled using the concept of solids of revolution. In this case, the tank is formed ...

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Calculus

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Applications of Integration

KEY TERMS AND CONCEPTS

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 x-axis. Each slice is a disk, and its radius is the height of the function at that point. The area of each disk is based on π times the square of the radius.

These thin disks together approximate the full tank volume. To find the total volume, their areas are added by integration along the length of the tank. After squaring the function that describes the shape, the expression becomes a constant times x squared times the difference between 2 and x. Expanding it gives terms with x to the third and x to the fourth powers.

Integrating these terms gives an antiderivative for the volume buildup. Evaluating the definite integral from 0 to 2 meters gives the final result. After simplification, the volume is about 1 cubic meter. This total internal capacity matters in aerospace engineering because engineers need accurate fuel capacity, weight distribution, and aircraft performance estimates.

Finding the Volume of a Jet Fuel Tank

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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...

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key terms and concepts

Use the disk method to find the volume of a jet fuel tank modeled as a solid of revolution. See how integration gives an answer of about 1 cubic meter.

Finding the Volume of a Jet Fuel Tank | 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.

Finding the Volume of a Jet Fuel Tank

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