Arc length
Problem 5.49 · medium
Find the length of the curve \( \displaystyle y = \frac{x^{4}}{8} + \frac{1}{4 x^{2}} \) from \( \displaystyle x = 1 \) to \( \displaystyle x = 3 \).
- Arc length is ∫ √(1 + (dy/dx)²) dx.
- \[ \frac{d}{d x} \left(\frac{x^{4}}{8} + \frac{1}{4 x^{2}}\right) = \frac{x^{3}}{2} - \frac{1}{2 x^{3}} \]dy/dx.✓ Proved
- \[ \left(\frac{x^{3}}{2} - \frac{1}{2 x^{3}}\right)^{2} + 1 = \frac{\left(x^{2} + 1\right)^{2} \left(x^{4} - x^{2} + 1\right)^{2}}{4 x^{6}} \]1 + (dy/dx)² simplifies (here to a perfect square, which is why these are set).✓ Proved
- \[ \int\limits_{1}^{3} \sqrt{\left(\frac{x^{3}}{2} - \frac{1}{2 x^{3}}\right)^{2} + 1}\, dx = \frac{92}{9} \]Integrate.✓ Proved
Answer \( \frac{92}{9} \)
Lines: 3 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. The explanation has not been reviewed yet.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Not checked | — | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | a 200,000-segment polygon along the curve has the same length |
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
generator:structured/arc_length, checked 2026-09-26 with SymPy 1.14.0.