Applications of Integrals · Arc Length & Surface Area
Arc Length
The length of a curve y = f(x) from x = a to x = b.
- 2 variables
- 2 worked examples
- 14 in Applications of Integrals
Variables
| Symbol | Name | Unit |
|---|---|---|
| a | Left bound | - |
| b | Right bound | - |
Worked examples
- f'(x) = (3/2)x^(1/2). [f']² = (9/4)x
- L = ∫₀⁴ √(1 + 9x/4) dx. Let u = 1+9x/4, du = 9/4 dx
- = (4/9)·(2/3)[u^(3/2)]₁¹⁰ = (8/27)(10√10 - 1)
Answer: (8/27)(10√10 - 1) ≈ 9.073
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Common questions
What is the Arc Length?
The length of a curve y = f(x) from x = a to x = b. It is one of the applications of integrals formulas in the CalcRef reference.
What do the symbols in the Arc Length mean?
In the Arc Length, a is the left bound; b is the right bound.
How do you use the Arc Length?
Worked example. Find the arc length of y = x^(3/2) from x = 0 to x = 4. f'(x) = (3/2)x^(1/2). [f']² = (9/4)x. L = ∫₀⁴ √(1 + 9x/4) dx. Let u = 1+9x/4, du = 9/4 dx. = (4/9)·(2/3)[u^(3/2)]₁¹⁰ = (8/27)(10√10 - 1). Answer: (8/27)(10√10 - 1) ≈ 9.073.
Practice
Quiz yourself on Applications of Integrals
Arc Length 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.