Calculus I · Unit 3B · lesson

Surface Area of Revolution

Concept

Learning objectives

Construct surface-area integrals from circumference times arc-length elements.

Surface Area of Revolution

Explanation

Surface area combines circumference with slanted length

A short piece of curve has length approximately $ds$ . Rotating it around an axis creates a narrow band whose area is approximately circumference times slanted width, $2\pi r\,ds$ . Substituting the arc-length element gives the standard surface-area formula. This explains why surface area includes both a radius factor and a square-root derivative factor.

The radius must be measured from the chosen axis, and the curve must be described without double-counting the surface. Surface-area integrals are sensitive to setup and may be difficult to evaluate exactly. A complete solution should identify the radius, the arc-length factor, the interval, and the units. Comparing simple cases with known cone or cylinder formulas is an excellent check on the model.

Rotating a short arc element around an axis produces a thin frustum-like band. Around the $x$ -axis,

S=2\pi\int_a^bf(x)\sqrt{1+[f'(x)]^2}\,dx,

provided $f(x)\ge0$ . The radius must be measured from the axis of rotation.

Worked example

A cone from a line segment

Rotate $y=x$ , $0\le x\le1$ , around the $x$ -axis:

S=2\pi\int_0^1x\sqrt2\,dx=\pi\sqrt2.

This is the lateral surface area of a cone with radius $1$ and slant height $\sqrt2$ .

Interactive checku3b-surface-01

Find the surface area generated by rotating $y=x$ , $0\le x\le1$ , around the x-axis.

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Show hint

Use radius x and arc factor $\sqrt2$ .

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After the explanation

Use the section idea

Reading lens

Distinguish geometric size from weighted amount: arc and surface formulas stretch local distance, while density assigns unequal mass to equal pieces.

Mental model

Arc length adds short chord lengths; surface area adds narrow bands; mass adds density-times-size; moments add position-weighted mass.

Decision

Identify the local size element first, then multiply by circumference, density, or position only when the modeled quantity requires it.

Common trap

Using geometric midpoint for a nonuniform object, forgetting the surface radius, or treating differential legs as independent finite lengths.

Check yourself

Can you explain why every factor belongs in the slice contribution and why the result has the intended units?

Source & rights

Original instruction with traceable references.

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