Problem 61 - Calculus Single Variable

Question

Calculate the area \(S\) of the surface obtained when the graph of the given function is rotated about the \(x\) -axis. $$ f(x)=\frac{3 x^{4}+1}{6 x} \quad 1 \leq x \leq 2 $$

Step-by-Step Solution

Answer

Set up the surface area integral, simplify, and evaluate from 1 to 2.

Step 1: Understand the Formula for Surface Area

To find the surface area of revolution about the x-axis, you use the formula: \[ S = 2\pi \int_{a}^{b} f(x) \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \ dx \] where \(f(x)\) is the function being rotated, and \(\frac{dy}{dx}\) is its derivative. In this problem, \( f(x) = \frac{3x^4 + 1}{6x} \) and the interval is \([1, 2]\).

Step 2: Find the Derivative of the Function

First, compute the derivative \( f'(x) \) of the given function. Let \( y = \frac{3x^4 + 1}{6x} \). Using the quotient rule, \( f'(x) = \frac{6x(12x^3) - (3x^4 + 1)(6)}{(6x)^2} \). Simplify this expression to find the derivative.

Step 3: Simplify the Derivative

Calculate \( f'(x) = \frac{72x^4 - 18x^4 - 6}{36x^2} \). Simplify to get \( f'(x) = \frac{54x^4 - 6}{36x^2} = \frac{9x^4 - 1}{6x^2} \).

Step 4: Set up the Surface Area Integral

Use the formula: \[ S = 2\pi \int_{1}^{2} f(x) \sqrt{1 + (f'(x))^2} \, dx \] Substituting the given function and its derivative gives: \[ S = 2\pi \int_{1}^{2} \left( \frac{3x^4 + 1}{6x} \right) \sqrt{1 + \left(\frac{9x^4 - 1}{6x^2}\right)^2} \ dx \].

Step 5: Compute \((f'(x))^2\) and the Radicand

Calculate \((f'(x))^2 = \left(\frac{9x^4 - 1}{6x^2}\right)^2 = \frac{(9x^4 - 1)^2}{36x^4} \). Plug this back into the square root: \( \sqrt{1 + \frac{(9x^4 - 1)^2}{36x^4}} \).

Step 6: Simplify the Integral Expression

The integral becomes: \[ S = 2\pi \int_{1}^{2} \left( \frac{3x^4 + 1}{6x} \right) \sqrt{1 + \frac{81x^8 - 18x^4 + 1}{36x^4}} \ dx \]. Simplify inside the square root and continue to solve the integral by standard methods or numerical approximation.

Step 7: Evaluate the Integral

Solve the definite integral from \(x = 1\) to \(x = 2\), either by analytical methods (if a simple antiderivative can be found) or using numerical methods like Simpson's Rule or a numerical integration tool. This will give you the area \(S\).

Step 8: Calculate the Final Result

Once the integral is evaluated, multiply the resulting value by \(2\pi\) to find the surface area of the solid of revolution.

Key Concepts

Calculus

Calculus is a branch of mathematics focused on change and motion. It consists of two main parts: differential calculus and integral calculus.
Differential calculus is concerned with rates of change such as when determining the slope of a curve. Integral calculus, on the other hand, deals with the accumulation of quantities. For example, it helps us find the area under curves, which is involved in calculating the surface area of revolution.
The process of calculus makes it possible to analyze problems involving continuously changing quantities and complex shapes. Understanding calculus is fundamental for solving problems like the surface area of a rotated curve. Its ability to describe real-world phenomena in mathematical terms is one of its many strengths.

Integration Techniques

Integration techniques are essential tools in calculus used to find the integral of a function, which is needed for computing areas and volumes, like the surface area of rotation. One crucial technique is the use of formulas or rules that simplify the integration process.

• Substitution: This helps simplify integrals by changing variables.
• Integration by Parts: It uses the product rule for differentiation in reverse.
• Trigonometric Substitution: Effective for integrals with square roots.

After setting up the integral for the surface area, these techniques may be applied for simplification. Often, when facing complex expressions, numerical integration methods like the trapezoidal rule are also considered for approximation. Understanding various integration techniques simplifies finding the antiderivative needed to evaluate definite integrals.

Derivative Calculation

Derivative calculation is a critical component of calculus, particularly for determining the slope or the rate of change of a function. When dealing with the surface area of a revolving curve, finding the derivative is crucial to incorporate into the surface area formula.
To calculate the derivative of a function like \[ y = \frac{3x^4 + 1}{6x} \], one method is the quotient rule. The quotient rule is used when differentiating a function that is the ratio of two other functions. The derivative of \( y \) with respect to \( x \) involves differentiating both the numerator and the denominator separately, and then applying the formula: \[ \left[\frac{u}{v}\right]' = \frac{v u' - u v'}{v^2} \]This yields the slope of the function at any given point, which is essential in calculating the expression inside the integral. Mastery of derivative calculation is vital for exploring dynamic systems and improving comprehension of calculus applications.

Definite Integral

A definite integral is an integral expressed between specific limits, symbolized as \( \int_{a}^{b} f(x) \, dx \). It provides the area under the curve of a function from \( x=a \) to \( x=b \), making it pivotal for calculating surface areas of revolved bodies.
When setting up the definite integral for the surface of revolution, it needs to include the function and its derivative for precise calculation. Solving the definite integral involves evaluating the integral first and then applying these bounds.
This integral gives the exact measure of accumulated change between two points, essential for exact calculations in physical and engineering processes. Knowing how to solve definite integrals, whether through analytical or numerical methods, enables greater flexibility and accuracy in handling diverse mathematical tasks.