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Area between curves

Problem 5.72 · medium Video

Find the area of the region bounded by \( \displaystyle y = - 3 x^{2} - 2 x - 3 \) and \( \displaystyle y = x - 3 \).

\[ - 3 x \left(x + 1\right) = - 3 x^{2} - 3 x \]
Set the curves equal: top − bottom, factored.✓ Proved
They cross at x = -1 and x = 0; between them y = -3*x**2 - 2*x - 3 is on top.
\[ \int\limits_{-1}^{0} \left(- 3 x^{2} - 3 x\right)\, dx = \frac{1}{2} \]
Area = ∫ (top − bottom) dx.✓ Proved

Answer \( \frac{1}{2} \)

The curves y = -3x² - 2x - 3 and y = x - 3 between x = -1 and x = 0. Thin rectangles fill the gap between them, getting narrower until they fill the region exactly; its area is 1/2.

Lines: 2 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt. How we verify

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

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — The intersection points are incorrect; solving -3x^2 - 2x - 3 = x - 3 yields x = 0 and x = -1, but the difference function is -3x^2 - 3x, which has roots at 0 and -1. Wait, let's recheck the intersection. -3x^2 - 2x - 3 = x - 3 => -3x^2 - 3x = 0 => -3x(x+1) = 0. So x=0 and x=-1 are correct. The difference is (-3x^2 - 2x - 3) - (x - 3) = -3x^2 - 3x. The integral of -3x^2 - 3x from -1 to 0 is [-x^3 - 1.5x^2] from -1 to 0 = 0 - (-(-1) - 1.5(1)) = - (1 - 1.5) = -(-0.5) = 0.5. The calculation is actually correct. Why did I think it was wrong? Let me re-read carefully. Equation 1: -3*x*(x + 1) is the factored form of -3x^2 - 3x. Correct. Sentence 2: Cross at -1 and 0. Correct. Top curve is parabola. Correct. Equation 3: Integral of -3x^2 - 3x from -1 to 0 is 1/2. Correct. Wait, is there a subtle error? Problem: y = -3x^2 - 2x - 3 and y = x - 3. Difference: (-3x^2 - 2x - 3) - (x - 3) = -3x^2 - 3x. Roots: -3x(x+1) = 0 -> x=0, x=-1. Integral from -1 to 0 of (-3x^2 - 3x) dx. Antiderivative: -x^3 - (3/2)x^2. At 0: 0. At -1: -(-1)^3 - (3/2)(-1)^2 = 1 - 1.5 = -0.5. Result: 0 - (-0.5) = 0.5. The solution seems perfectly correct. Why would it fail? Let me check the prompt again. "Find the area of the region bounded by...". Is it possible the curves don't bound a region? No, they intersect at two points and one is above the other between them. Is the arithmetic wrong? No. Is the setup wrong? No. Let me look closer at Sentence 2: "They cross at x = -1 and x = 0; between them y = -3*x**2 - 2*x - 3 is on top." Let's test a point between -1 and 0, e.g., x = -0.5. Parabola: -3(0.25) - 2(-0.5) - 3 = -0.75 + 1 - 3 = -2.75. Line: -0.5 - 3 = -3.5. -2.75 > -3.5. So the parabola is on top. Correct. Is there any ambiguity? No. Maybe the "proved" tag on equation 1 is misleading? It says "-3*x*(x + 1), -3*x**2 - 3*x". This is just an identity. Maybe the integral limits? -1 to 0. Correct. I will mark it as pass.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-09-28 — The intersection points are incorrect; solving -3x^2 - 2x - 3 = x - 3 yields x = 0 and x = -1, but the difference function is -3x^2 - 3x, which has roots at 0 and -1. Wait, let's recheck the intersection. -3x^2 - 2x - 3 = x - 3 => -3x^2 - 3x = 0 => -3x(x+1) = 0. So x=0 and x=-1 are correct. The difference is (-3x^2 - 2x - 3) - (x - 3) = -3x^2 - 3x. The integral of -3x^2 - 3x from -1 to 0 is [-x^3 - 1.5x^2] from -1 to 0 = 0 - (-(-1) - 1.5(1)) = - (1 - 1.5) = -(-0.5) = 0.5. The calculation is actually correct. Why did I think it was wrong? Let me re-read carefully. Equation 1: -3*x*(x + 1) is the factored form of -3x^2 - 3x. Correct. Sentence 2: Cross at -1 and 0. Correct. Top curve is parabola. Correct. Equation 3: Integral of -3x^2 - 3x from -1 to 0 is 1/2. Correct. Wait, is there a subtle error? Problem: y = -3x^2 - 2x - 3 and y = x - 3. Difference: (-3x^2 - 2x - 3) - (x - 3) = -3x^2 - 3x. Roots: -3x(x+1) = 0 -> x=0, x=-1. Integral from -1 to 0 of (-3x^2 - 3x) dx. Antiderivative: -x^3 - (3/2)x^2. At 0: 0. At -1: -(-1)^3 - (3/2)(-1)^2 = 1 - 1.5 = -0.5. Result: 0 - (-0.5) = 0.5. The solution seems perfectly correct. Why would it fail? Let me check the prompt again. "Find the area of the region bounded by...". Is it possible the curves don't bound a region? No, they intersect at two points and one is above the other between them. Is the arithmetic wrong? No. Is the setup wrong? No. Let me look closer at Sentence 2: "They cross at x = -1 and x = 0; between them y = -3*x**2 - 2*x - 3 is on top." Let's test a point between -1 and 0, e.g., x = -0.5. Parabola: -3(0.25) - 2(-0.5) - 3 = -0.75 + 1 - 3 = -2.75. Line: -0.5 - 3 = -3.5. -2.75 > -3.5. So the parabola is on top. Correct. Is there any ambiguity? No. Maybe the "proved" tag on equation 1 is misleading? It says "-3*x*(x + 1), -3*x**2 - 3*x". This is just an identity. Maybe the integral limits? -1 to 0. Correct. I will mark it as pass.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: inconclusive 2026-09-27 — reviewer returned a non-object
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The intersection points are incorrect; solving -3x^2 - 2x - 3 = x - 3 yields x = 0 and x = -1, but substituting x = -1 into the original equations gives y = -2 for the parabola and y = -4 for the line, so they do not intersect at x = -1. The correct intersections are at x = 0 and x = -1/3? No, -3x^2 - 3x = 0 => x=0, -1. Wait, let's re-solve: -3x^2 - 2x - 3 = x - 3 => -3x^2 - 3x = 0 => -3x(x+1)=0. So x=0 and x=-1 are indeed the intersection points. My previous thought was wrong. Let's re-verify the y-values. At x=-1: y_parabola = -3(1) - 2(-1) - 3 = -3 + 2 - 3 = -4. y_line = -1 - 3 = -4. They do intersect. So the bounds are correct. Is the top/bottom correct? At x = -0.5: y_parabola = -3(0.25) - 2(-0.5) - 3 = -0.75 + 1 - 3 = -2.75. y_line = -0.5 - 3 = -3.5. Parabola is higher. So top/bottom is correct. The integral calculation: int(-3x^2 - 3x) from -1 to 0. Antiderivative: -x^3 - 1.5x^2. Eval at 0: 0. Eval at -1: -(-1) - 1.5(1) = 1 - 1.5 = -0.5. Result: 0 - (-0.5) = 0.5. The solution is actually correct. Why did I think it was wrong? I misread the problem or my mental math failed. Let me double check the problem statement again. y = -3x^2 - 2x - 3 and y = x - 3. Difference: (-3x^2 - 2x - 3) - (x - 3) = -3x^2 - 3x. Roots at 0, -1. Integral is 1/2. The solution is correct.
gpt-oss:20b: pass 2026-09-26

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 2026-09-28 with SymPy 1.14.0.