Problem 21 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

A circus tent has cylindrical symmetry about its center pole. The height a distance of \(x\) feet from the center pole is given by \(h(x)=\frac{8}{1+\frac{x^{2}}{16}}\) feet. What is the volume enclosed by the tent of radius \(4 ?\)

Step-by-Step Solution

Answer

The volume enclosed by the tent of radius 4 is \(32\pi\) cubic feet.

Step 1: Revision on volume of a solid of revolution

The volume of a solid of revolution about the x-axis, from \(x=a\) to \(x=b\), is given by the formula \(V=\pi \int_{a}^{b}[f(x)]^{2} dx\). This is called the disc method. In this case, \(f(x)=h(x)\) gives the height of the tent as a function of \(x\), the distance from the center.

Step 2: Substituting the function into the formula

We need to find the volume of the tent of radius 4, so \(a=0\) and \(b=4\). Plug \(f(x) = h(x) = \frac{8}{1+\frac{x^{2}}{16}}\) into the formula for \(V\), to get \(V = \pi \int_{0}^{4} [\frac{8}{1+\frac{x^{2}}{16}}]^{2} dx\).

Step 3: Compute the integral to find the volume

To compute the integral, we square the function to get \(\frac{64}{(1+\frac{x^{2}}{16})^{2}}\). Now we can solve the integral and multiply the result by \(\pi\) to get the volume. The integral is a standard integral and can be referred from integral tables. Solving it and multiplying it by \(\pi\), we get that the volume is \(\pi*[32*\arctan(4/4)-32*\arctan(0)]=32\pi\) cubic feet.

Key Concepts

Disc Method

When we talk about finding the volume of a solid of revolution, the disc method is a crucial tool. Imagine slicing a solid into thin discs or circles, similar to slicing a loaf of bread. Each disc has a small thickness and a certain radius, based on the function you are working with. By stacking up these discs, you can approximate the volume of the solid.
The formula to calculate the volume using the disc method is:

• \(V=\pi \int_{a}^{b}[f(x)]^{2} \ dx\)

Where:

• \(V\) is the volume of the solid,
• \([f(x)]^2\) is the area of each disc,
• \([a, b]\) is the interval over which you are calculating the volume.

It's important to understand that by squaring the function \(f(x)\), you find the area of the circular faces of the discs. This squared value represents the radius squared times \(\pi\), calculating the area for each disc in the solid.

Integral Computation

Performing integral computation is essential in finding the volume of a solid when using the disc method. Think of integration as a process to accumulate all the infinite small areas (discs) into one whole volume.
To compute the integral for the tent example, you begin by plugging the function \(h(x)\) into the formula:

• \(V = \pi \int_{0}^{4} \left[\frac{8}{1+\frac{x^{2}}{16}}\right]^2 \ dx\)

Once you substitute the function into the integral, the next step involves squaring \(h(x)\) and simplifying the expression inside the integral.

• You obtain \(\frac{64}{(1+\frac{x^2}{16})^2}\) after squaring.

This is a standard integral that typically requires referencing an integral table or using known integration techniques. After solving the integral, multiplying by \(\pi\) finalizes your computation, giving you the desired volume. In this exercise, the integral evaluates to give a precise result, ensuring full comprehension of the solid's dimensions.

Solid of Revolution

A solid of revolution is formed when a two-dimensional shape spins around an axis. This axis acts like a skewer that the shape twirls around, forming a three-dimensional object. Consider rotating a simple curve or line around an axis; the shape morphs into a symmetrical solid.
In our tent example, the height expressed by \(h(x)\) is revolved around the x-axis. This action creates the enclosed volume we are trying to measure, mimicking the shape of the circus tent.
Understanding a solid of revolution helps in visualizing the effects of this rotation:

• The process creates a solid with cylindrical symmetry.
• It demonstrates how the initial shape expands in three dimensions.
• Understanding this concept is vital when applying methods like the disc or shell method for volume calculation.

By connecting these shapes and formations, learners can easily imagine how simple functions lead to complex three-dimensional solids through revolution.