Area between curves
Problem 5.1 · medium
Find the area of the region bounded by \( \displaystyle y = - x^{2} + 4 x + 3 \) and \( \displaystyle y = 3 x + 1 \).
- \[ \left(2 - x\right) \left(x + 1\right) = - x^{2} + x + 2 \]Set the curves equal: top − bottom, factored.✓ Proved
- They cross at x = -1 and x = 2; between them y = -x**2 + 4*x + 3 is on top.
- \[ \int\limits_{-1}^{2} \left(- x^{2} + x + 2\right)\, dx = \frac{9}{2} \]Area = ∫ (top − bottom) dx.✓ Proved
Answer \( \frac{9}{2} \)
Lines: 2 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 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | Not checked | — | a sentence; read, not computed |
| 3 | ✓ 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 | numerical quadrature of |top − bottom| gives the same area |
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/area_between_curves, checked 2026-09-26 with SymPy 1.14.0.