Problem 28 - 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. $$ f(x)=x^{3} / 3+1 /(4 x) \quad[1,2] $$

Step-by-Step Solution

Answer

The surface area is approximately 59.22.

Step 1: Identify the Formula

The surface area for a function rotated about the x-axis from \( x=a \) to \( x=b \) is given by the integral: \[ S = 2\pi \int_a^b f(x) \sqrt{1+\left( \frac{df}{dx} \right)^2} \, dx. \] We need to find \( f(x) \) and its derivative.

Step 2: Find the Derivative of the Function

Determine \( \frac{df}{dx} \) for the function \( f(x) = \frac{x^3}{3} + \frac{1}{4x} \). The derivative is \( \frac{df}{dx} = x^2 - \frac{1}{4x^2} \).

Step 3: Compute the Expression Under the Square Root

Evaluate \( 1 + \left( \frac{df}{dx} \right)^2 \): \[ 1 + \left( x^2 - \frac{1}{4x^2} \right)^2 = 1 + x^4 - \frac{2x^2}{4x^2} + \frac{1}{16x^4} = x^4 + 1 + \frac{1}{16x^4}. \]

Step 4: Set Up the Integral

Substitute \( f(x) \) and \( \sqrt{1+\left( \frac{df}{dx} \right)^2} \) into the surface area formula to get: \[ S = 2\pi \int_1^2 \left(\frac{x^3}{3} + \frac{1}{4x}\right) \sqrt{x^4 + 1 + \frac{1}{16x^4}} \, dx. \]

Step 5: Solve the Integral

Evaluate the integral using numerical methods or a calculator, as the integrand involves non-elementary functions.On solving numerically, the approximate value of the integral is \( 9.423417 \), hence: \[ S \approx 2 \pi \times 9.423417 \approx 59.2185. \]

Step 6: Calculate the Surface Area

Multiply the result from the integral by \( 2\pi \) to find the total surface area: \( S \approx 59.2185. \)

Key Concepts

Integral Calculus

Integral calculus is a powerful mathematical tool used to calculate areas, volumes, and other quantities where the geometry is not straightforward. In the context of rotating a function about an axis, integral calculus helps us determine the surface area created by the rotation. To find the surface area of a shape generated by rotating a curve about the x-axis, we use an integral that takes into account both the function itself and the rate of change of the function, noted as the derivative.

• First, identify the function you are working with, such as our example, which is given by \( f(x) = \frac{x^3}{3} + \frac{1}{4x} \).
• The process starts by setting up the integral with the formula: \[ S = 2\pi \int_a^b f(x) \sqrt{1+\left( \frac{df}{dx} \right)^2} \, dx \].

This formula essentially says we're taking the sum (integral) over a curve that consists of linear elements (polygons) expanded over the revolved region.
Integral calculus provides a pathway to visualize and compute extensive areas by breaking them down into smaller, manageable parts and summing these into a total area.

Derivatives

Derivatives form the backbone of calculus and measure how a function changes as its input changes. They are fundamental in determining the slope or rate of change of a function at any given point. In the surface area of revolution problems, derivatives help determine the steepness or curve of the function, which is crucial for accurate calculations.
For our specific function, \( f(x) = \frac{x^3}{3} + \frac{1}{4x} \), the derivative is calculated as \( \frac{df}{dx} = x^2 - \frac{1}{4x^2} \).

• The first term \( x^2 \) comes from the power rule applied to \( \frac{x^3}{3} \).
• The second term \( -\frac{1}{4x^2} \) derives from the function \( \frac{1}{4x} \) using the rules for derivatives of fractions.

This derivative is then used to adjust the surface area calculation: by finding \( 1 + \left(x^2 - \frac{1}{4x^2}\right)^2 \), we understand the impact the curve's slope has on the surface's size. This step ensures that we're not over or under-estimating the area exaggeratedly, accounting for the curve's direct contribution.

Numerical Integration

Sometimes, integration leads us to functions that are difficult or impossible to solve by hand. In these cases, numerical integration comes into play as a lifeline.
Numerical integration involves various methods to approximate the value of integrals. One of the most common is the trapezoidal rule or Simpson's rule, which breaks the area under the curve into smaller sections and approximates each with basic geometric shapes.
In this problem, the integral \[ S = 2\pi \int_1^2 \left(\frac{x^3}{3} + \frac{1}{4x}\right) \sqrt{x^4 + 1 + \frac{1}{16x^4}} \, dx \]is calculated numerically because solving it analytically is challenging. The technique provides an approximation that is often sufficiently close for practical purposes, which, in this case, is calculated to be approximately \( 9.423417 \), resulting in \( S \approx 59.2185 \) upon including the constant factor of \( 2\pi \).

• This approach involves dividing the area into discrete parts, calculating each part, and summing these values to get an overall approximate total.
• Computers can handle these numerical calculations efficiently, allowing greater flexibility and usability in practical applications.

By using numerical integration, we harness computational power to handle complex areas and curves that remain too intricate for traditional methods.