Problem 34 - Calculus and its applications

Question

A regulation football used in the National Football League (NFL) is 11 in. from tip to tip and 7 in. in diameter at its thickest (the regulations allow for slight variation in these dimensions). (Source: NFL.) The shape of a football can be modeled by the function $$ f(x)=-0.116 x^{2}+3.5, \quad \text { for }-5.5 \leq x \leq 5.5 $$ where \(x\) is in inches. Find the volume of an NFL football by rotating the area bounded by the graph of \(f\) around the \(x\) -axis.

Step-by-Step Solution

Answer

The volume of the NFL football is approximately 553 cubic inches.

Step 1: Understand the Problem

We need to find the volume of the football, which is modeled mathematically. The function given represents half of the cross-sectional profile of the football. We determine the volume by rotating this curve 360 degrees around the x-axis.

Step 2: Determine the Volume Formula

The volume of a solid of revolution created by revolving a function around the x-axis is given by the integral: \[ V = \int_{a}^{b} \pi [f(x)]^2 \, dx \] For this problem, \(a = -5.5\) and \(b = 5.5\), and \(f(x) = -0.116x^2 + 3.5\).

Step 3: Set Up the Integral

Substitute the function \(f(x)\) into the volume formula: \[ V = \int_{-5.5}^{5.5} \pi (-0.116x^2 + 3.5)^2 \, dx \] This integral represents the volume of the football when the area under \(f(x)\) is revolved around the x-axis.

Step 4: Expand the Function Inside the Integral

First, square the function: \[ (-0.116x^2 + 3.5)^2 = (0.116)^2x^4 - 2 \times 0.116 \times 3.5 x^2 + 3.5^2 \] Simplifying this gives: \[ (0.013456)x^4 - (0.812)x^2 + 12.25 \]

Step 5: Evaluate the Integral

Now substitute back into the integral: \[ V = \pi \int_{-5.5}^{5.5} (0.013456x^4 - 0.812x^2 + 12.25) \, dx \] Calculate the integral term-by-term: \[ V = \pi ([\frac{0.013456}{5}x^5] - [\frac{0.812}{3}x^3] + [12.25x])] \,\bigg|^{-5.5}_{5.5}\] Evaluate this to find the volume.

Step 6: Substitute and Simplify the Solution

Calculate: - First, plug \(x = 5.5\) into all terms, and then \(x = -5.5\). - Subtract the negative area values from the positive area values. After simplification, this provides the final numerical answer for the volume.

Key Concepts

Volume of Rotation

The volume of rotation is a fascinating concept in integral calculus where we calculate the volume of a solid formed by revolving a two-dimensional shape around an axis. In this particular exercise, we have a curve defined by the function that mimics the half-profile of a football. By revolving this half-profile around the x-axis, we obtain the full three-dimensional shape of the football. Imagine taking a flat shape and spinning it around a line; it creates a solid object. When we talk about volume of rotation, we refer to the space that this object occupies.

The foundation of calculating this volume lies in integral calculus. Specifically, we use the method of integrating the area under the curve, squared, and then multiplied by \(\pi\). This method essentially sums up all the circular slices along the axis into one cohesive volume, almost like adding up the thickness of multiple disks that stack up to form the whole shape. Calculating volume this way gives us a precise way of determining the amount of space a rotating object encompasses.

Solid of Revolution

A solid of revolution is a solid figure obtained by rotating a plane curve around a straight line (the axis of revolution) that lies on the same plane. In the exercise, the solid of revolution is the football, modeled by a specific mathematical function. The graph of this function creates the shape which, once revolved around the x-axis, becomes a three-dimensional football.

Understanding how to construct a solid of revolution helps in a variety of real-world applications, from designing objects in engineering to determining physical properties like volume and surface area. When dealing with solids of revolution in calculus, we frequently rely on methods like the disk or washer method, based on the object’s symmetry and the axis around which it is rotated.

The disk method is appropriate when the solid is made by revolving the area between a single curve and the x-axis, which is precisely the case with this football model. To visualize, imagine each point on the curve generating a circle of a different radius when rotated; together, these circles form the solid.

Integral Calculus

Integral calculus is the branch of mathematics that deals with finding the total size, length, area, or volume of objects. It achieves this by accumulation, which is crucial for problems involving continuous change, like calculating the volume of a solid of revolution.

In this problem, we use an integral to handle the accumulation of volumes from infinitesimally thin slices (or disks) along the length of the football. We set up an integral from \(-5.5\) to \(5.5\) which represents the entire length of the football from one tip to the other. The function \(f(x)\) that defines the football is squared and further integrated, a process governed by laws of calculus, to capture the continuous volume generated by rotating around the x-axis.

Integral calculus is essential here because it allows us to translate the geometric phenomenon of a rotating two-dimensional shape into a manageable calculation. Through integration, we convert its infinite processes into a single tangible result—the volume of the football.