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Definite integrals from area formulas

Problem 4.1369 · easy Video

Evaluate \( \displaystyle \int_{-1}^{4} 2 x + 2 \, dx \) using a formula from geometry.

The region between the graph and the x-axis is a trapezoid (a triangle, since one side has height 0).
Reviewed
\[ \frac{\left(0 + 10\right) \left(1 + 4\right)}{2} = 25 \]
Its (signed) area.✓ Proved
\[ \int\limits_{-1}^{4} \left(2 x + 2\right)\, dx = 25 \]
Integrating directly gives the same.✓ Proved

Answer \( 25 \)

The graph of f(x) = 2x + 2 from x = -1 to x = 4. Rectangles under the curve get thinner until they fill the area, 25.

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify

The full receipt

Line	Status	Checked by	Detail
1	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	numerical quadrature

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the region as a trapezoid (degenerate to a triangle at the lower bound) and applies the geometric area formula accurately. The logic is sound and the result is correct.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-29 — The solution correctly identifies the region as a trapezoid (degenerate to a triangle at the lower bound) and applies the geometric area formula accurately. The logic is sound and the result is correct.
gpt-oss:20b: pass 2026-10-29
qwen3.6:27b-mlx: pass 2026-10-28 — The solution correctly identifies the region as a trapezoid (with one vertical side of length 0, effectively a triangle) and applies the correct area formula. The bounds and function values are handled correctly.
gpt-oss:20b: pass 2026-10-28
qwen3.6:27b-mlx: pass 2026-10-28 — The solution correctly identifies the region as a trapezoid (specifically a right triangle since the height at x=-1 is 0) and applies the geometric area formula correctly to obtain the result.
gpt-oss:20b: pass 2026-10-28

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/integral_geometry, checked 2026-10-29 with SymPy 1.14.0.