Applications of Integrals · Arc Length & Surface Area
Surface Area of Revolution
The surface area generated by revolving y = f(x) about the x-axis.
- 2 variables
- 2 worked examples
- 14 in Applications of Integrals
Variables
| Symbol | Name | Unit |
|---|---|---|
| a | Left bound | - |
| b | Right bound | - |
Worked examples
- f'(x) = 1. √(1+1) = √2
- S = 2π ∫₀¹ x · √2 dx = 2π√2 [x²/2]₀¹ = π√2
Answer: π√2 ≈ 4.443
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Common questions
What is the Surface Area of Revolution?
The surface area generated by revolving y = f(x) about the x-axis. It is one of the applications of integrals formulas in the CalcRef reference.
When does the Surface Area of Revolution apply?
The Surface Area of Revolution holds under this condition: f(x) ≥ 0 on [a, b].
How do you use the Surface Area of Revolution?
Worked example. Find the surface area when y = x on [0, 1] is revolved about the x-axis. f'(x) = 1. √(1+1) = √2. S = 2π ∫₀¹ x · √2 dx = 2π√2 [x²/2]₀¹ = π√2. Answer: π√2 ≈ 4.443.
Practice
Quiz yourself on Applications of Integrals
Surface Area of Revolution is one of 14 Applications of Integrals formulas in CalcRef, out of 136 in the library. Today's free round is 10 mixed multiple-choice questions drawn from the same dataset as this page, refreshed every day and replayable as often as you like - no account needed.