Final answer:
To find the surface area of the solid formed by rotating the curve y = x²/4 - ln(x/2) about the x-axis from x = 1 to x = 2, we must compute the integral of 2π times the function times the square root of 1 plus the square of its derivative, all with respect to x.
Step-by-step explanation:
The question involves finding the exact surface area of a solid formed by revolving a curve about an axis, which is a common problem in calculus. To find this area, we need to set up and evaluate a surface area integral. For the curve y = x²/4 - ln(x/2), rotated about the x-axis from x = 1 to x = 2, we use the formula for the surface area of a solid of revolution, which is an integral of the form:
S = 2π ∫_a^b f(x) ∙ √(1 + (f'(x))²) dx
First, we need to calculate the derivative of the function f'(x) which is f'(x) = x/2 - 1/(x). Then we need to find the square of the derivative, (f'(x))² and add 1 to it to get 1 + (f'(x))². After finding the integral of 2π ∫_1^2 (x²/4 - ln(x/2)) ∙ √(1 + (x/2 - 1/(x))²) dx, we will compute it without the use of technology. It involves basic integration techniques such as substitution, integration by parts, and perhaps partial fractions for the logarithmic part of the function.
By integrating this expression from 1 to 2, we obtain the exact area of the surface created by the rotation of the given curve about the x-axis.