The area of the surface formed by revolving a curve $y = f(x)$ about the x-axis is given by the formula:

$A = 2\pi \int_{a}^{b} y \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \,dx$

Step 1: Compute $\frac{dy}{dx}$

Given:

$y = \frac{x^3}{12} + \frac{1}{x}$

Differentiate $y$ with respect to $x$:

$\frac{dy}{dx} = \frac{d}{dx} \left( \frac{x^3}{12} + \frac{1}{x} \right)$

$\frac{dy}{dx} = \frac{3x^2}{12} - \frac{1}{x^2}$

$\frac{dy}{dx} = \frac{x^2}{4} - \frac{1}{x^2}$

Step 2: Compute $1 + \left(\frac{dy}{dx}\right)^2$

$\left(\frac{dy}{dx}\right)^2 = \left(\frac{x^2}{4} - \frac{1}{x^2} \right)^2$

Expanding the square:

$\left(\frac{x^2}{4} - \frac{1}{x^2} \right)^2 = \frac{x^4}{16} - 2\frac{x^2}{4} \frac{1}{x^2} + \frac{1}{x^4}$

$= \frac{x^4}{16} - \frac{2}{4} + \frac{1}{x^4}$

$= \frac{x^4}{16} - \frac{1}{2} + \frac{1}{x^4}$

Thus,

$1 + \left(\frac{dy}{dx}\right)^2 = 1 + \frac{x^4}{16} - \frac{1}{2} + \frac{1}{x^4}$

$= \frac{16}{16} + \frac{x^4}{16} - \frac{8}{16} + \frac{1}{x^4}$

$= \frac{x^4 + 8 + 16/x^4}{16}$

$= \frac{x^4 + 8 + 16/x^4}{16}$

$= \frac{(x^2 - 4/x^2)^2 + 16}{16}$

Taking the square root:

$\sqrt{1 + \left(\frac{dy}{dx}\right)^2} = \frac{\sqrt{x^4 + 8 + 16/x^4}}{4}$

Step 3: Set Up the Integral

$A = 2\pi \int_{0.7}^{1.8} \left(\frac{x^3}{12} + \frac{1}{x}\right) \frac{\sqrt{x^4 + 8 + 16/x^4}}{4} dx$

This integral does not have a simple elementary antiderivative and is best solved numerically.

I will compute the numerical approximation. scipy.integrate as spi import numpy as np

Define the function to integrate

def surface_area_function(x): y = (x3 / 12) + (1 / x) dy_dx = (x2 / 4) - (1 / x2) sqrt_term = np.sqrt(1 + dy_dx2) return 2 * np.pi * y * sqrt_term / 4

Compute the numerical integral

a, b = 0.7, 1.8 surface_area, error = spi.quad(surface_area_function, a, b)

surface_areaThe approximate surface area of the surface formed by revolving the curve about the x-axis is 2.265 square units.

Would you like a step-by-step breakdown of the numerical integration or any additional details?

Related Questions:

1. How do we derive the formula for surface area of revolution?
2. What are some real-world applications of surface area of revolution?
3. Can we use parametric or polar equations for surface area calculations?
4. How does the surface area change if the function is rotated about the y-axis instead?
5. How can we approximate the integral using Simpson’s Rule?

Tip:

When dealing with integrals that do not have elementary solutions, numerical methods like Simpson's Rule, Trapezoidal Rule, or Quadrature are useful for approximations.