Problem 40 - Thomas Calculus in SI Units

Question

Find the volumes of the solids generated by revolving the regions bounded by the lines and curves about the \(x\) -axis. $$y=\sec x, \quad y=\tan x, \quad x=0, \quad x=1$$

Step-by-Step Solution

Answer

The volume is \(\pi\).

Step 1: Understand the Problem

The problem involves finding the volume of the solid formed by rotating the region bounded by the given curves and lines around the x-axis. The curves are \(y = \sec x\) and \(y = \tan x\) between \(x = 0\) and \(x = 1\).

Step 2: Visualize the Region

Sketch the curves \(y = \sec x\) and \(y = \tan x\) and identify the region between \(x = 0\) and \(x = 1\). The area between these functions and relative to the x-axis forms our region of interest.

Step 3: Use Washer Method Formula

The volume \(V\) of the solid is given by the formula for the washer method: \[ V = \pi \int_{a}^{b} [(R(x))^2 - (r(x))^2] \, dx \]where \(R(x)\) is the outer radius and \(r(x)\) is the inner radius of washers perpendicular to the x-axis. In this case, \(R(x) = \sec x\) and \(r(x) = \tan x\).

Step 4: Set Up the Integral

The integral becomes:\[ V = \pi \int_{0}^{1} [ (\sec x)^2 - (\tan x)^2 ] \, dx \]

Step 5: Simplify the Integrand

Using the identity \((\sec x)^2 - (\tan x)^2 = 1\), simplify the integral to:\[ V = \pi \int_{0}^{1} 1 \, dx \]

Step 6: Compute the Integral

Evaluate the integral:\[ V = \pi [ x ]_{0}^{1} = \pi (1 - 0) = \pi \]

Step 7: State the Result

The volume of the solid generated by revolving the region about the x-axis is \(\pi\).

Key Concepts

Washer Method

The washer method is an essential technique in calculus for finding the volume of solids of revolution. This method is especially useful when the solid has a hole in its center, much like a washer. The technique involves slicing the solid perpendicularly to the axis of revolution and calculating the volume of each washer-shaped slice.

The process starts with identifying the outer and inner radii of the washer. If you visualize the slices, the outer edge forms a circle with radius \(R(x)\), and the inner edge forms a circle with radius \(r(x)\). The difference between their areas gives the cross-sectional area of the washer. The formula used is:

• Volume \(V = \pi \int_{a}^{b} [(R(x))^2 - (r(x))^2] \, dx \)

where \(R(x)\) is the radius of the outer function, and \(r(x)\) is the radius of the inner function. By integrating this area across the bounds \(a\) and \(b\), we find the total volume of the solid.

Integral Calculus

Integral calculus is a fundamental branch of mathematical analysis dealing with the integration of functions. It focuses on finding the total size, value, or outcome over a given interval. Integrals are widely used to compute quantities like areas, volumes, and other accumulative properties.

In the context of this exercise, integral calculus is used to sum up the infinite number of infinitesimally thin washers to find the total volume of the solid. The key aspect is setting up the integral correctly by determining the functions \(R(x)\) and \(r(x)\) that represent the boundaries of the rotation area. Here, the integral calculates the sum of these washer volumes from \(x = 0\) to \(x = 1\).

Understanding how to translate the physical problem into a mathematical integral is crucial in solving complex geometry problems with precision.

Trigonometric Functions

Trigonometric functions are functions of an angle used throughout mathematics to relate the angles and sides of right triangles. In this exercise, the focus lies on \(y = \sec x\) and \(y = \tan x\), which are trigonometric functions derived from the basic parameters of a triangle.

• \(\sec x\) represents the secant of \(x\), which is the reciprocal of the cosine function \((\sec x = 1/\cos x)\).
• \(\tan x\) denotes the tangent of \(x\), expressed as \(\sin x / \cos x\).

Trigonometric functions have magnificent applications in studying periodic phenomena and, as seen here, in determining curves for solid rotations.

They help model the curves of \(y = \sec x\) and \(y = \tan x\), defining the region between \(x=0\) and \(x=1\) that we revolve around the x-axis for volume calculations.

Revolution Around X-axis

Revolution around the x-axis is a technique used to generate three-dimensional shapes by rotating a two-dimensional region about the horizontal axis. This method often leads to the creation of symmetrical solids, which are easier to analyze and compute.

When we revolve a region about the x-axis, every point in this region traces a circular path, forming a solid. In our case, the region bounded by \(y = \sec x\), \(y = \tan x\), \(x = 0\), and \(x = 1\) forms a solid by this revolution. The Washer Method is then applied to find the volume of this newly formed solid.

Mathematically, the problem transitions from a two-dimensional shape into a solid object whose volume can be calculated by understanding the path of rotation and corresponding radii of each point within the given range. Using calculus to calculate these volumes gives insights into the structure and dimensions of the solid.