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Calculus IIChapter 2: Applications of IntegralsSection 2-2

Surface Area — Question 7

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Question 7

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

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Original worksheet page 1: question and worked solution for 2-2-007
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Question 7 – Solution

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Step 1: Differentiate and set up the improper integral. y′=−1x2.y'=-\frac1{x^2}. Thus S=2π∫1∞1x1+1x4dx.S=2\pi\int_1^\infty\frac1x\sqrt{1+\frac1{x^4}}\,dx. Step 2: Find a lower comparison. Since 1+x−4>1,\sqrt{1+x^{-4}}>1, we have 1x1+x−4>1x.\frac1x\sqrt{1+x^{-4}}>\frac1x. Step 3: Test the comparison integral. ∫1∞dxx=limb→∞ln⁡b=∞.\int_1^\infty\frac{dx}{x} =\lim_{b\to\infty}\ln b=\infty. By direct comparison, the surface-area integral diverges. S=∞ (infinite surface area)\boxed{S=\infty\text{ (infinite surface area)}}

Original worksheet page 2: question and worked solution for 2-2-007

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