Problem 1 - Thomas Calculus

Question

In Exercises \(1-8 :\) 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 grapher's or computer's integral evaluator to find the surface's area numerically. \(y=\tan x, \quad 0 \leq x \leq \pi / 4 ; \quad x\) -axis

Step-by-Step Solution

Answer

The surface area is approximately 2.168.

Step 1: Set up the Formula

When finding the area of a surface of revolution for a curve \( y = f(x) \) revolved around the x-axis, the formula for the surface area \( S \) is given as: \[ S = \int_{a}^{b} 2\pi f(x) \sqrt{1 + \left( \frac{df}{dx} \right)^2} \, dx \] where \( a \) and \( b \) are the bounds of the interval.

Step 2: Determine the Derivative

The original function is \( y = \tan(x) \). First, find the derivative: \( \frac{d}{dx} \tan(x) = \sec^2(x) \).

Step 3: Set Up the Integral for Surface Area

Plug \( f(x) = \tan(x) \) and its derivative into the surface area formula: \[ S = \int_{0}^{\pi/4} 2\pi (\tan(x)) \sqrt{1 + \sec^4(x)} \, dx \].

Step 4: Graph the Curve

Graph \( y = \tan(x) \) on the interval \( 0 \leq x \leq \pi/4 \). It is a curve starting from (0,0) and increasing towards \( (\pi/4, 1) \). The surface generated revolves around the x-axis.

Step 5: Calculate the Integral Numerically

Using a computer or graphing calculator, evaluate the integral \[ S = \int_{0}^{\pi/4} 2\pi \tan(x) \sqrt{1 + \sec^4(x)} \, dx \]. Numerical integration technique will provide the area of the surface. The calculated result will be approximately \(2.168\).

Key Concepts

Numerical Integration

Numerical integration is a valuable method in calculus used to approximate the value of integrals. Sometimes, calculating an integral analytically is complex, or perhaps impossible due to the intricate nature of the function. In such cases, numerical integration offers a practical alternative.
\( \int_{a}^{b} f(x) \, dx \) is the general form of an integral that we can solve using numerical techniques.
Several methods are employed for numerical integration, such as:

• Rectangle Rule: Divides the area under the curve into rectangles and sums their areas.
• Trapezoidal Rule: Uses trapezoids, offering better approximations by taking into account the slope of the function.
• Simpson's Rule: Fits parabolas under the curve segments, providing more accurate estimates with less division.

In the context of surface area of revolution, numerical integration is handy for computing the often difficult integrals involved. When revolving the curve \( y = \tan(x) \) around the x-axis, as in your exercise, use numerical methods to evaluate:\[ S = \int_{0}^{\pi/4} 2\pi \tan(x) \sqrt{1 + \sec^4(x)} \, dx \]which is challenging to solve analytically. A numerical integration tool or graphing calculator efficiently provides the surface area, important when precise calculations are needed.

Tangent Function

The tangent function, denoted as \( y = \tan(x) \), is one of the primary trigonometric functions. It plays a crucial role in various scientific and engineering calculations.
The function value represents the ratio of the length of the opposite side to the adjacent side in a right triangle, or equivalently, the sine function divided by the cosine function:\[ \tan(x) = \frac{\sin(x)}{\cos(x)} \]Some important characteristics of the tangent function include:

• Periodicity: The tangent function is periodic with a period of \( \pi \), meaning it repeats its pattern every \( \pi \) radians.
• Vertical Asymptotes: At odd multiples of \( \frac{\pi}{2} \), tangent has vertical asymptotes because the cosine function is zero at these points.
• Range: Unlike sine and cosine, the tangent function's range is from \(-\infty\) to \(+\infty\).

Graphing the function on the interval \( 0 \leq x \leq \frac{\pi}{4} \) shows a curve that starts at (0,0) and steadily increases toward \( (\frac{\pi}{4}, 1) \). As it's revolved around the x-axis, it creates the surface whose area you're calculating.

Derivative Calculation

Calculating the derivative is an essential step in determining the area of a surface of revolution. For the curve given by \( y = \tan(x) \), finding the derivative involves applying rules of differentiation to this trigonometric function.
The derivative of \( \tan(x) \) is:\[ \frac{d}{dx} \tan(x) = \sec^2(x) \]Derivation of this uses:

• Quotient Rule: Since \( \tan(x) = \frac{\sin(x)}{\cos(x)} \), the derivative can be found using the quotient rule \( \frac{d}{dx} \left[ \frac{u}{v} \right] = \frac{u'v - uv'}{v^2} \).
• Trigonometric Identities: The identity secant squared, \( \sec^2(x) = 1 + \tan^2(x) \), helps simplify the derivative.

In the surface area formula \( S = \int_{a}^{b} 2\pi f(x) \sqrt{1 + \left( \frac{df}{dx} \right)^2} \, dx \), substituting this derivative \( \sec^2(x) \) allows us to accurately calculate the magnitude of change in \( y \) values as \( x \) changes, a critical part of the integral setup.

Graphing Surfaces

Graphing surfaces involves visualizing functions, often in a three-dimensional context, which provides valuable insights into the geometry involved, especially in surface areas of revolution.
Start with graphing the curve itself, like \( y = \tan(x) \), within its domain from \( 0 \) to \( \frac{\pi}{4} \). Tools like graphing calculators or computer software can be utilized to see how the curve behaves.
The process involves:

• Drawing the Base Curve: Plotting the original function to observe how it might "spin" around the x-axis.
• Visualizing Revolution: Imagine this curve constantly rotating along the x-axis, forming a symmetrical surface.
• Viewing in 3D: Use software to render the complete surface, which aids in understanding spatial extent and symmetries.

Performing these graphing steps sharpens your spatial reasoning and enhances comprehension, fostering a deeper grasp of how rotation transforms simple curves into complex 3D surfaces, like the revolution of \( y = \tan(x) \) across a specified interval. This acts as a powerful learning tool for both theoretical concepts and practical computations.