Problem 1 - Thomas Calculus in SI Units

Question

a. Set up an integral for the area of the surface generated by revolving the given curve about the indicated axis. b. Graph the curve to see what it looks like. If you can, graph the surface too. c. Use your utility's integral evaluator to find the surface's area numerically. $$y=\tan x, \quad 0 \leq x \leq \pi / 4 ; \quad x \text { -axis }$$

Step-by-Step Solution

Answer

The surface area is found by evaluating \( S = \int_0^{\pi/4} 2\pi \tan x \sqrt{1 + \sec^4 x}\,dx \) numerically.

Step 1: Understand the Problem Statement

We need to find the surface area generated by revolving the curve \( y = \tan x \) around the x-axis over the interval \( 0 \leq x \leq \pi/4 \). To do this, we'll set up the integral for surface area using the formula for surface area of revolution around the x-axis.

Step 2: Set Up the Integral for Surface Area

The formula for the surface area \( S \) of a curve \( y = f(x) \) revolved around the x-axis from \( x = a \) to \( x = b \) is \( S = \int_a^b 2\pi f(x)\sqrt{1 + (f'(x))^2}\,dx \). Here, \( f(x) = \tan x \) and we need to find \( f'(x) \).

Step 3: Compute the Derivative

Find the derivative of \( f(x) = \tan x \). The derivative is \( f'(x) = \sec^2 x \).

Step 4: Substitute into the Surface Area Formula

Substitute \( f(x) = \tan x \) and \( f'(x) = \sec^2 x \) into the formula:\[ S = \int_0^{\pi/4} 2\pi \tan x \sqrt{1 + \sec^4 x}\,dx \].

Step 5: Graph the Curve

To visualize \( y = \tan x \) from \( 0 \) to \( \pi/4 \), we can create a plot showing that it's an increasing curve from \( y = 0 \) to \( y = 1 \). The surface of revolution will look like a trumpet bell or frustum if revolved around the x-axis.

Step 6: Evaluate the Integral Numerically

Use a numerical integrator, like a graphing calculator or software tool, to evaluate the definite integral:\[ S = \int_0^{\pi/4} 2\pi \tan x \sqrt{1 + \sec^4 x}\,dx \]. The numeric result will approximate the surface area.

Key Concepts

Integration

Integration is a fundamental concept in calculus. Here, it helps us find the area of a surface created by revolving a curve about an axis. The process involves breaking up a problem into small calculable parts, adding them up to derive a total value. In this exercise, we need to compute the surface area of revolution using integration.

The general formula for the surface area of a curve revolved around the x-axis is given by: \[ S = \int_a^b 2\pi f(x)\sqrt{1 + (f'(x))^2}\,dx \]Where:

• a and b are the bounds of integration, in this case, 0 and \( \pi/4 \)
• \( f(x) \) is the function being revolved, here \( \tan x \)
• \( f'(x) \) is the derivative of the function, \( \sec^2 x \) for \( \tan x \)

Through integration, we sum the infinite number of infinitely small rings to find the total surface area.

Derivative

The derivative is a key tool in calculus used to determine the rate at which one quantity changes with respect to another. In the context of a surface of revolution, the derivative helps in accounting for the slope of the curve being revolved.

For the function \( y = \tan x \), the derivative is calculated as \( f'(x) = \sec^2 x \).
This reveals how fast \( y \) changes with each tiny increment along \( x \). The derivative provides the crucial part \( \sqrt{1 + (f'(x))^2} \) in the surface area formula.

The squared secant term, \( \sec^4 x \), modifies the curve's contribution to the overall shape when revolved, impacting the total surface area.

Graphing

Graphing the curve can offer valuable insights into its behavior, making it easier to understand how the surface will look once revolved. In this exercise, we graph \( y = \tan x \) within the interval \( 0 \leq x \leq \pi/4 \).

During this range, we observe that \( \tan x \) starts at 0 and increases to 1. This gradual increase indicates that the curve steadily climbs and won't form any sharp turns or loops.

Upon revolving this curve around the x-axis, expect to see a smooth, bell-like shape resembling a trumpet. This visual representation makes it easier to comprehend the integration solution.

Numerical Integration

Numerical integration is employed when analytical solutions seem complex or impractical. It approximates the value of integrals using various techniques and is particularly useful for functions that do not yield straightforward solutions.

After formulating the integral \[ S = \int_0^{\pi/4} 2\pi \tan x \sqrt{1 + \sec^4 x}\,dx \]we may use numerical methods, like the trapezoidal rule or Simpson's rule, allowing us to estimate the exact surface area.

Graphing calculators or software like MATLAB can handle the calculations, providing a quick way to obtain practical answers when hand calculations become cumbersome.
Understanding numerical integration ensures we can approach situations requiring practical solutions efficiently, complementing our theoretical knowledge.