Problem 24 - Thomas Calculus

Question

Find the volumes of the solids generated by revolving the regions bounded by the lines and curves about the \(x\) -axis. \begin{equation} y=\sec x, \quad y=0, \quad x=-\pi / 4, \quad x=\pi / 4 \end{equation}

Step-by-Step Solution

Answer

The volume of the solid is \( 2\pi \).

Step 1: Understand the Problem

We need to find the volume of the solid formed by revolving the region bounded by the curves \( y = \sec x \), \( y = 0 \), and the lines \( x = -\frac{\pi}{4} \) and \( x = \frac{\pi}{4} \) about the \( x \)-axis.

Step 2: Identify the Integral Setup

We will use the disk method to find the volume of the solid. The volume \( V \) is given by the integral formula: \[ V = \pi \int_{a}^{b} [f(x)]^2 \, dx \]where \( f(x) \) is \( \sec x \), and \( a = -\frac{\pi}{4} \), \( b = \frac{\pi}{4} \).

Step 3: Write the Integral

Using the disk method's formula, write the integral to find the volume of the solid:\[ V = \pi \int_{-\pi/4}^{\pi/4} (\sec x)^2 \, dx \].

Step 4: Solve the Integral

To solve the integral, remember that the antiderivative of \( \sec^2 x \) is \( \tan x \). Thus, the integral becomes:\[ V = \pi [\tan x]_{-\pi/4}^{\pi/4} \].

Step 5: Calculate the Integral

Evaluate the definite integral:\[ V = \pi [\tan(\pi/4) - \tan(-\pi/4)] \].Since \( \tan(\pi/4) = 1 \) and \( \tan(-\pi/4) = -1 \), this simplifies to:\[ V = \pi [1 - (-1)] = \pi \times 2 = 2\pi \].

Step 6: Conclusion

The volume of the solid formed by the revolution of the described region about the \( x \)-axis is \( 2\pi \).

Key Concepts

Disk Method

The disk method is a technique used in calculus for finding the volume of a solid of revolution. This method involves slicing the solid perpendicular to the axis of rotation, creating disk-shaped cross-sections. Here's how it works:

• First, identify the region that needs to be revolved around an axis. In our exercise, it's the area between the curve, the x-axis, and the specified vertical lines.
• Then, picture slicing this region into thin slabs, perpendicular to the x-axis. Each of these slices forms a disk when the region is revolved.
• The radius of each disk is determined by the distance from the x-axis to the curve, denoted by the function value at each x, which is \( f(x) \) or \( \sec x \).
• The volume of each infinitesimally thin disk is approximated as \( \pi [f(x)]^2 \, dx \).
• Summing the volumes of these disks (integrating) from \( x = a \) to \( x = b \) gives the total volume.

Using this method, the integral formula becomes \( V = \pi \int_{a}^{b} [f(x)]^2 \, dx \), which allows us to calculate the volume through integration.

Definite Integral

The definite integral is crucial in calculating areas and volumes in calculus. It represents the net area under a curve between two set limits, known as the bounds of integration. In our exercise, here’s how it’s applied:

• We seek the definite integral from \( x = -\pi/4 \) to \( x = \pi/4 \). This gives the total accumulation of values of \( (\sec x)^2 \) across the interval.


• The concept of definite integral involves summing infinitesimally small quantities represented by the integral sign \( \int \).
• It combines these small quantities to account for a whole, like the total volume of a solid in our problem.
• When we evaluate a definite integral, as in \( \pi [\tan x]_{-\pi/4}^{\pi/4} \), it involves calculating the antiderivative at the upper limit and subtracting the antiderivative evaluated at the lower limit.

This process helps us exactly determine the volume of the rotated solid by considering every single point within the bounds, thereby getting a precise measurement.

Revolved Solids

Revolved solids are created when a shape or region is rotated around an axis, forming a three-dimensional object. The process involves:

• Identifying a two-dimensional area that will be revolved. In this exercise, the area between the curve \( y = \sec x \), the line \( y=0 \), and the limits \( x = -\pi/4 \) to \( x=\pi/4 \) is revolved around the x-axis.
• The revolution transforms this 2D region into a symmetrical 3D solid, akin to a vase that's been spun around a horizontal rod.
• These 3D objects can have various cross-sectional shapes, depending on how the original 2D shape is bounded and rotated.
• The disk method is often used to calculate the volume of such solids by evaluating the accumulation of these cross-sections through integration.

This solid of revolution concept showcases the application of calculus in visualizing and calculating the properties of complex shapes, demonstrating how a simple integral can convert two-dimensional areas into tangible three-dimensional volumes. This method provides deeper insight into spatial reasoning and its mathematical representation.