Problem 6 - Thomas Calculus

Question

Find the volumes of the solids. 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

The volume expressions require evaluating respective integrals for both shapes.

Step 1: Understand the Problem

We need to find the volumes of two different solids. The cross-sections change depending on the part specified: (a) circular disks, and (b) squares. These cross sections range from the curve \(y=\tan x\) to \(y=\sec x\), within the limits \(x=\frac{-\pi}{3}\) to \(x=\frac{\pi}{3}\).

Step 2: Calculate Diameter of Circular Disks

The diameter of the circular disk is the distance between the curves \(y=\sec x\) and \(y=\tan x\). Therefore, the diameter \(d\) is given by \(d = \sec x - \tan x\). Radius \(r\) is \(r = \frac{d}{2} = \frac{\sec x - \tan x}{2}\).

Step 3: Volume Calculation of Circular Disks

The area \(A\) of each circular cross-section can be expressed using the formula \(A = \pi r^2 = \pi \left( \frac{\sec x - \tan x}{2} \right)^2\). To find the volume \(V\), integrate this area from \(x = \frac{-\pi}{3}\) to \(x = \frac{\pi}{3}\): \[ V = \int_{-\pi/3}^{\pi/3} \pi \left( \frac{\sec x - \tan x}{2} \right)^2 \mathrm{d}x \]

Step 4: Volume Expression for Circular Disks

Calculate the above integral to determine the volume expressed in terms of \(x\): \[ V = \frac{\pi}{4} \int_{-\pi/3}^{\pi/3} (\sec x - \tan x)^2 \mathrm{d}x \] Use trigonometric identities and integral tables to solve this.

Step 5: Calculate Side of Squares

For the squares, the side length \(s\) of each square is also the distance between the curves: \(s = \sec x - \tan x\).

Step 6: Volume Calculation of Squares

The area \(A\) of each square cross-section is \(A = s^2 = (\sec x - \tan x)^2\). To find the volume \(V\), integrate this area: \[ V = \int_{-\pi/3}^{\pi/3} (\sec x - \tan x)^2 \mathrm{d}x \]

Step 7: Volume Expression for Squares

Compute the integral \(\int_{-\pi/3}^{\pi/3} (\sec x - \tan x)^2 \mathrm{d}x\) using similar methods as with circular disks, considering the trigonometric identities.

Key Concepts

Cross Sections

Cross sections are foundational in understanding the geometry of solids. Imagine slicing the solid at different points along the axis. Each slice or cross section gives you a consistent shape, like a cross-section of a loaf of bread. In this exercise, our cross-sections are either circular disks or squares. These slices run perpendicular to the x-axis, meaning they're vertical slices standing upright.

• The cross section for a circular disk spans the distance from the curve \(y = \tan x\) to \(y = \sec x\).
• For squares, the base's length is the same span as the circular disk's diameter, \(y = \tan x\) to \(y = \sec x\).

Understanding how these cross-sections change as we move along the x-axis helps in setting up the integrals to find volumes.

Definite Integrals

Definite integrals play a crucial role in calculating the volume of solids with variable cross-sections. In this problem, we integrate the area of each cross section as it varies along the x-axis.The integral captures the idea of summing up the infinite cross sections from one limit to another. Here, our limits go from \(x = -\pi / 3\) to \(x = \pi / 3\).For the circular disk:- We use the integral of \(\pi r^2\), where \(r = \frac{\sec x - \tan x}{2}\).For the square:- The integral involves \(s^2\), where \(s\) is the length \(\sec x - \tan x\).By integrating these areas with respect to \(x\), we determine the complete volume of the solids.

Trigonometric Curves

The curves \(y = \tan x\) and \(y = \sec x\) define the boundaries of our cross-sections.

• \( y = \tan x \): This curve represents the tangent of \(x\), a trigonometric function that blossoms towards infinity as \(x\) approaches \( \pi/2 \).
• \( y = \sec x \): This is the secant of \(x\), which is the reciprocal of the cosine function, similarly infinite at \(x = \pi/2\) and undefined where \(\cos x = 0\).

These curves intersect the plane at certain angles, creating varying widths for the circular and square cross-sections, thereby influencing the volume calculations.

Circular Disks

The concept of circular disks simplifies into geometry and calculus. Calculating the volume of a solid with circular disks involves more than just geometry – integration is key.Each disk's diameter stretches to be the distance between the curves \(y = \tan x\) and \(y = \sec x\). As such, the disk’s radius is half this length.To envision:- Radius \(r\) is \( \frac{\sec x - \tan x}{2} \).- To find the volume, integrate the area of the disks from \(x = -\pi/3\) to \(x=\pi/3\).The definite integral needed is:\[V = \frac{\pi}{4} \int_{-\pi/3}^{\pi/3} (\sec x - \tan x)^2 \, \mathrm{d}x\]This slices the volume into every possible disk and combines them into the solid's total volume.

Squares

Square cross-sections are straightforward yet powerful in forming the foundation of the solid’s volume calculation.For these squares, their side length is the distance between our two curves \(y = \tan x\) to \(y = \sec x\).

• Side length \(s\) equals \(\sec x - \tan x\).
• The area of a square being \(s^2\) directly gives us the shape's slice through the solid.

To find the volume using these square cross-sections, integrate the area formula for squares:\[V = \int_{-\pi/3}^{\pi/3} (\sec x - \tan x)^2 \, \mathrm{d}x\]This integral sums up slices of square area, providing the composite volume of the entire solid.