Surface Area — Question 7

Question 7

Curve: $y=1/x$ on $x\ge1$.
Axis of rotation: the $x$-axis.
Task: Determine whether the generated surface has finite area.

Question 7 – Solution

Step 1: Differentiate and set up the improper integral. $$y'=-\frac1{x^2}.$$ Thus $$S=2\pi\int_1^\infty\frac1x\sqrt{1+\frac1{x^4}}\,dx.$$ Step 2: Find a lower comparison. Since $$\sqrt{1+x^{-4}}>1,$$ we have $$\frac1x\sqrt{1+x^{-4}}>\frac1x.$$ Step 3: Test the comparison integral. $$\int_1^\infty\frac{dx}{x} =\lim_{b\to\infty}\ln b=\infty.$$ By direct comparison, the surface-area integral diverges. $$\boxed{S=\infty\text{ (infinite surface area)}}$$