Arc length
Problem 5.59 · hard
Find the length of the curve \( \displaystyle y = \frac{2 \left(x + 2\right)^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 0 \) to \( \displaystyle x = 7 \).
- Arc length is ∫ √(1 + (dy/dx)²) dx.
- \[ \frac{d}{d x} \frac{2 \left(x + 2\right)^{\frac{3}{2}}}{3} = \sqrt{x + 2} \]dy/dx.✓ Proved
- \[ x + 3 \]1 + (dy/dx)² simplifies (here to a perfect square, which is why these are set).✓ Proved
- \[ \int\limits_{0}^{7} \sqrt{x + 3}\, dx = - 2 \sqrt{3} + \frac{20 \sqrt{10}}{3} \]Integrate.✓ Proved
Answer \( - 2 \sqrt{3} + \frac{20 \sqrt{10}}{3} \)
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.