Problem 8 - Thomas Calculus

Question

Find the volumes of the solids in Exercises \(3-10\) . The solid lies between planes perpendicular to the \(x\) -axis at \(x=-\pi / 3\) and \(x=\pi / 3 .\) The cross-sections perpendicular to the \(x\) -axis are a. circular disks with diameters running from the curve \(y=\tan x\) to the curve \(y=\sec x .\) b. squares whose bases run from the curve \(y=\tan x\) to the curve \(y=\sec x\)

Step-by-Step Solution

Answer

Compute and integrate the cross-section areas: disks and squares.

Step 1: Identify the Cross-Section Area for Circular Disks

The cross-sections are circular disks. The diameter of each disk is the distance between the curves \(y = \tan x\) and \(y = \sec x\). The length of this diameter is given by \(\sec x - \tan x\). Therefore, the radius \(r\) of each disk is \(r = \frac{\sec x - \tan x}{2}\). The area \(A\) of each disk's cross-section is: \[ A = \pi \left(\frac{\sec x - \tan x}{2}\right)^2 \].

Step 2: Setup the Integral for Volume of Circular Disks

The volume \(V\) of the solid is found by integrating the area of the cross-section from the lower limit \(-\pi/3\) to the upper limit \(\pi/3\). Thus, the volume is given by: \[ V = \int_{-\pi/3}^{\pi/3} \pi \left(\frac{\sec x - \tan x}{2}\right)^2 \, dx \].

Step 3: Evaluate the Integral for Volume of Circular Disks

Evaluate the integral to find the volume. This can be done analytically or using numerical integration.\[ V = \int_{-\pi/3}^{\pi/3} \pi \left(\frac{\sec x - \tan x}{2}\right)^2 \, dx \approx \pi \left(\frac{1}{2}\right)^2 \int_{-\pi/3}^{\pi/3} (\sec x - \tan x )^2 \, dx \]. Integrate and simplify to get the final volume.

Step 4: Identify the Cross-Section Area for Squares

For the square cross sections, the side length is the distance between \(y = \tan x\) and \(y = \sec x\), which is \(\sec x - \tan x\). The area \(A\) of each square is:\[ A = (\sec x - \tan x)^2 \].

Step 5: Setup the Integral for Volume of Squares

The volume \(V\) of the solid is obtained by integrating the area of the square cross-sections from \(-\pi/3\) to \(\pi/3\):\[ V = \int_{-\pi/3}^{\pi/3} (\sec x - \tan x)^2 \, dx \].

Step 6: Evaluate the Integral for Volume of Squares

Evaluate the integral for the squares to find the volume. Perform the integration:\[ V = \int_{-\pi/3}^{\pi/3} (\sec x - \tan x)^2 \, dx \]. Integrate and simplify to get the final volume.

Key Concepts

Volume of Solids

Understanding the volume of solids is essential in calculus. It involves calculating how much space an object occupies in three-dimensional space. Finding the volume of a solid can be approached in various ways depending on its shape and structure. For instance, in this exercise, the volume of a solid is determined by understanding the cross-sectional areas and integrating these areas along a specified range.

In our example, the solid is bounded by planes perpendicular to the x-axis from \(x = -\pi/3\) to \(x = \pi/3\). The volume here is calculated by finding the area of each cross-section and integrating this area across the interval. This type of problem often requires comprehensive knowledge of geometry and calculus to set up the correct integral and derive the volume.

Integration

Integration is a crucial concept in calculus, especially when calculating areas and volumes. It can be seen as the reverse process of differentiation. In this context, integration helps in summing up infinitely small elements to find the total volume of a solid.

To find the volume of the solid with circular disk cross-sections, we first determine the area of a single disk at a point \(x\). The disk's radius is derived from the difference between the functions \(y = \sec x\) and \(y = \tan x\). This radius leads to the calculation of the disk’s area, which is then integrated from \(x = -\pi/3\) to \(x = \pi/3\) to find the full volume.

• The integral is: \[ V = \int_{-\pi/3}^{\pi/3} \pi \left(\frac{\sec x - \tan x}{2}\right)^2 \, dx \]
• For square sections, the setup is: \[ V = \int_{-\pi/3}^{\pi/3} (\sec x - \tan x)^2 \, dx \]

Cross-sections

Cross-sections refer to the shapes you get when slicing a three-dimensional object perpendicular to a specified axis. They are crucial in calculating volumes because you can integrate over the area of these cross-sections to get the total volume.

In the given exercise, two types of cross-sections are considered:

• Circular Disks: These sections have diameters defined by the difference between \(y = \sec x\) and \(y = \tan x\). By halving this difference, you get the radius, which is then used to compute the area of the disk.
• Squares: Here, the base of the square is also \(\sec x - \tan x\). The area of each cross-section is simply the square of this base length.

By understanding and calculating the area of these cross-sections, we can set up the proper integrals to find the volume of the entire solid.

Trigonometric Functions

Trigonometric functions like \(\tan x\) and \(\sec x\) are important in this exercise. These functions describe the curves that define the boundaries of our solid’s cross-sections.

The tangent function \(\tan x\) is the ratio of the sine to cosine, while the secant \(\sec x\) is the reciprocal of the cosine. They are periodic functions, and their graphs show how they rise and fall over intervals.

• Given functions:
  - \(y = \tan x\)
  - \(y = \sec x\)
• Both functions help determine the shape and size of each cross-section.

The difference \(\sec x - \tan x\) gives us crucial measurements like diameter and side length for our cross-sections. A firm grasp of these trigonometric functions can significantly ease the process of solving such volume problems in calculus.