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 π 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’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.