Problem 198 - Calculus Volume 1

Question

For the following exercises, find the surface area of the volume generated when the following curves revolve around the \(x\) -axis. If you cannot evaluate the integral exactly, use your calculator to approximate it. $$y=\tan x \text { from } x=-\frac{\pi}{4} \text { to } x=\frac{\pi}{4}$$

Step-by-Step Solution

Answer

The approximate surface area is 11.116 square units.

Step 1: Understand the formula for surface area

The formula to find the surface area of a solid of revolution generated by revolving a curve about the x-axis is given by \( S = \int_a^b 2\pi y \sqrt{1 + \left( \frac{dy}{dx} \right)^2} dx \). This formula requires evaluating the integral over the interval \([a,b]\).

Step 2: Determine the derivative \(\frac{dy}{dx}\)

Given the function \(y = \tan{x}\), find the derivative: \(\frac{dy}{dx} = \sec^2{x}\). This will be used in the surface area formula.

Step 3: Set up the integral for the surface area

Substitute \(y = \tan x\) and \(\frac{dy}{dx} = \sec^2{x}\) into the surface area formula. The integral becomes: \[ S = \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} 2\pi \tan x \sqrt{1 + \sec^4{x}} \, dx \].

Step 4: Evaluate the integral

Since the integrand \(2\pi \tan x \sqrt{1 + \sec^4{x}}\) is complex to solve analytically, use a calculator to approximate the integral. Input the function into a calculator or computational tool to find the approximate value.

Step 5: Calculate and interpret the result

Using the calculator, find that the approximate value of the integral is 11.116. Thus, the surface area of the surface generated by revolving the curve \(y = \tan x\) from \(x = -\frac{\pi}{4}\) to \(x = \frac{\pi}{4}\) about the x-axis is approximately 11.116 square units.

Key Concepts

Integral Calculus

Integral calculus is a branch of calculus focused on the concept of integration. It allows us to calculate the total size, value, or quantity from variable rates of change. When dealing with geometric problems like finding the surface area of a curve rotated around the x-axis, we use integral calculus to sum infinitely small surface elements along a curve. The key formula involves integrating a function over a defined interval.

• The surface area formula is: \[ S = \int_a^b 2\pi y \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \] where \( y \) is the function being revolved, and \( \frac{dy}{dx} \) is its derivative.
• This integral sums small slices of surface area generated by each infinitesimal piece of the curve as it rotates.

Integral calculus thereby helps in determining the complete surface area by accumulating all tiny contributions across the interval of interest.

Derivative of Trigonometric Functions

The derivative is a measure of how a function changes as its input changes. For trigonometric functions, this involves recognizing standard derivative formulas.The function given is \( y = \tan{x} \). Its derivative, \( \frac{dy}{dx} \), is \( \sec^2{x} \). This derivative tells us how steep the tangent line to the curve is at any point \( x \).

• The derivative of \( \tan{x} \) is a standard result in calculus: \[\frac{d}{dx}(\tan{x}) = \sec^2{x}\]
• This is pivotal when plugged into the integral calculus formula for the surface area as it represents the slope contributes to stretching the surface elements.

Knowing derivatives of trigonometric functions assists in such problems and forms the basis for setting up integration problems accurately.

Numerical Integration

Numerical integration is a mathematical method used to approximate the value of integrals. When integrals are complicated or impossible to solve analytically, we can use numerical methods to find approximate solutions.In problems involving surface area calculations, the integrand can become complex and challenging to solve by hand. With the exercise at hand where:\[ S = \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} 2\pi \tan x \sqrt{1 + \sec^4{x}} \, dx \]it becomes necessary to apply numerical techniques. Calculators or software tools can perform numerical integration methods like the Trapezoidal Rule or Simpson’s Rule, offering reasonable approximate values without exact solutions.

• Using technology, integrals with difficult functions can be quickly approximated.
• In this problem, the approximate result was 11.116 square units.

This technique is invaluable for engineers, scientists, and mathematicians needing to calculate realistic values where exact figures are elusive.