How to Find Surface Area of a Revolved Curve on [a, b]

Math Problem Statement

Assume f is a nonnegative function with a continuous first derivative on [a, b]. The curve y = f(x) on [a, b] is revolved about the x-axis. Explain how to find the area of the surface that is generated.

Question content area bottom

Part 1

What is the area of the surface generated when the graph of f on the interval [a, b] is revolved about the x-axis?

To find the area of the surface that is generated, evaluate the integral

Upper S equals Integral from a to b pi f left parenthesis x right parenthesis StartRoot 1 plus f prime left parenthesis x right parenthesis EndRoot dxS=∫abπ f(x)1+f′(x)dx.

To find the area of the surface that is generated, evaluate the integral

Upper S equals Integral from a to b 2 pi f left parenthesis x right parenthesis StartRoot 1 plus left parenthesis f prime left parenthesis x right parenthesis right parenthesis squared EndRoot dxS=∫ab2π f(x)1+f′(x)2dx.

To find the area of the surface that is generated, evaluate the integral

Upper S equals Integral from a to b 2 pi StartRoot 1 plus left parenthesis f prime left parenthesis x right parenthesis right parenthesis squared EndRoot dxS=∫ab2π 1+f′(x)2dx.

To find the area of the surface that is generated, evaluate the integral

Upper S equals Integral from a to b pi f left parenthesis x right parenthesis StartRoot 1 plus left parenthesis f prime left parenthesis x right parenthesis right parenthesis squared EndRoot dxS=∫abπ f(x)1+f′(x)2dx.

Solution

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Math Problem Analysis

Mathematical Concepts

Surface Area of Revolution
Differentiation
Definite Integral
Arc Length

Formulas

Surface Area S = ∫[a,b] 2πf(x)√(1 + (f'(x))^2) dx

Theorems

Formula for Surface Area of Revolution
First Derivative Theorem

Suitable Grade Level

Grades 11-12, College Level Calculus

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