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Area between curves

Problem 5.170 · medium Video

Find the area of the region bounded by \( \displaystyle y = - 3 x^{2} - 6 x - 1 \) and \( \displaystyle y = 3 x - 1 \).

\[ - 3 x \left(x + 3\right) = - 3 x^{2} - 9 x \]
Set the curves equal: top − bottom, factored.✓ Proved
They cross at x = -3 and x = 0; between them y = -3*x**2 - 6*x - 1 is on top.
\[ \int\limits_{-3}^{0} \left(- 3 x^{2} - 9 x\right)\, dx = \frac{27}{2} \]
Area = ∫ (top − bottom) dx.✓ Proved

Answer \( \frac{27}{2} \)

The curves y = -3x² - 6x - 1 and y = 3x - 1 between x = -3 and x = 0. Thin rectangles fill the gap between them, getting narrower until they fill the region exactly; its area is 27/2.

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 integrand in step 1 is incorrect; the difference (top - bottom) is -3x^2 - 9x, but step 2 claims the top curve is y = -3x^2 - 6x - 1, which implies the difference should be (-3x^2 - 6x - 1) - (3x - 1) = -3x^2 - 9x. Wait, let me re-calculate. Top: -3x^2 - 6x - 1. Bottom: 3x - 1. Difference: (-3x^2 - 6x - 1) - (3x - 1) = -3x^2 - 9x. The algebra in step 1 is actually correct for the difference. However, step 2 says 'They cross at x = -3 and x = 0'. Let's check intersections: -3x^2 - 6x - 1 = 3x - 1 => -3x^2 - 9x = 0 => -3x(x+3) = 0. Roots are 0 and -3. This is correct. The integral of -3x^2 - 9x from -3 to 0 is [-x^3 - 4.5x^2] from -3 to 0. At 0: 0. At -3: -(-27) - 4.5(9) = 27 - 40.5 = -13.5. So 0 - (-13.5) = 13.5 = 27/2. The calculation is correct. Why did I think it was wrong? Let me re-read carefully. Step 1 equation: -3*x*(x + 3) = -3*x**2 - 9*x. This is the integrand. Step 2 sentence: 'They cross at x = -3 and x = 0; between them y = -3*x**2 - 6*x - 1 is on top.' This is correct. Step 3 equation: Integral(-3*x**2 - 9*x, (x, -3, 0)) = 27/2. This is correct. Is there a subtle error? The problem asks for the area bounded by the curves. The setup is correct. The answer is correct. Wait, look at step 1 again. It says '-- Set the curves equal: top − bottom, factored.' The equation shown is `-3*x*(x + 3) = -3*x**2 - 9*x`. This is an identity, not the setup of the integral. But the comment explains it. Is there a misleading part? No. Is there an error? No. Let me double check the 'top' claim. At x = -1, Top = -3(1) - 6(-1) - 1 = -3 + 6 - 1 = 2. Bottom = 3(-1) - 1 = -4. Top > Bottom. Correct. The solution seems sound. Why would it fail? Maybe the prompt implies I should find something wrong. Let me look really closely. Step 1: `-3*x*(x + 3) = -3*x**2 - 9*x`. This is just expanding the factored form. It doesn't explicitly state the integral limits or the function being integrated in the equation itself, but the comment does. Step 3 calculates the integral. The result is 27/2. The stated answer is 27/2. Everything seems correct. Is it possible the user made a typo in the problem statement or solution? No, I must review what is given. If it is correct, I pass. Wait, looking at step 1 again. The comment says 'Set the curves equal: top - bottom, factored.' The equation is just an algebraic identity. It doesn't show the subtraction. But step 3 uses the expanded form. This is a bit disjointed but not wrong. Let me check the severity definitions. 'Error' is a step or setup is wrong. 'Misleading' is a student would learn something false. 'Style' is correct but unclear. The jump from step 1 (an identity) to step 3 (an integral) skips stating the integral explicitly in a sentence, but step 3's comment says 'Area = ...'. This is acceptable. I will pass.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-04 — The integrand in step 1 is incorrect; the difference (top - bottom) is -3x^2 - 9x, but step 2 claims the top curve is y = -3x^2 - 6x - 1, which implies the difference should be (-3x^2 - 6x - 1) - (3x - 1) = -3x^2 - 9x. Wait, let me re-calculate. Top: -3x^2 - 6x - 1. Bottom: 3x - 1. Difference: (-3x^2 - 6x - 1) - (3x - 1) = -3x^2 - 9x. The algebra in step 1 is actually correct for the difference. However, step 2 says 'They cross at x = -3 and x = 0'. Let's check intersections: -3x^2 - 6x - 1 = 3x - 1 => -3x^2 - 9x = 0 => -3x(x+3) = 0. Roots are 0 and -3. This is correct. The integral of -3x^2 - 9x from -3 to 0 is [-x^3 - 4.5x^2] from -3 to 0. At 0: 0. At -3: -(-27) - 4.5(9) = 27 - 40.5 = -13.5. So 0 - (-13.5) = 13.5 = 27/2. The calculation is correct. Why did I think it was wrong? Let me re-read carefully. Step 1 equation: -3*x*(x + 3) = -3*x**2 - 9*x. This is the integrand. Step 2 sentence: 'They cross at x = -3 and x = 0; between them y = -3*x**2 - 6*x - 1 is on top.' This is correct. Step 3 equation: Integral(-3*x**2 - 9*x, (x, -3, 0)) = 27/2. This is correct. Is there a subtle error? The problem asks for the area bounded by the curves. The setup is correct. The answer is correct. Wait, look at step 1 again. It says '-- Set the curves equal: top − bottom, factored.' The equation shown is `-3*x*(x + 3) = -3*x**2 - 9*x`. This is an identity, not the setup of the integral. But the comment explains it. Is there a misleading part? No. Is there an error? No. Let me double check the 'top' claim. At x = -1, Top = -3(1) - 6(-1) - 1 = -3 + 6 - 1 = 2. Bottom = 3(-1) - 1 = -4. Top > Bottom. Correct. The solution seems sound. Why would it fail? Maybe the prompt implies I should find something wrong. Let me look really closely. Step 1: `-3*x*(x + 3) = -3*x**2 - 9*x`. This is just expanding the factored form. It doesn't explicitly state the integral limits or the function being integrated in the equation itself, but the comment does. Step 3 calculates the integral. The result is 27/2. The stated answer is 27/2. Everything seems correct. Is it possible the user made a typo in the problem statement or solution? No, I must review what is given. If it is correct, I pass. Wait, looking at step 1 again. The comment says 'Set the curves equal: top - bottom, factored.' The equation is just an algebraic identity. It doesn't show the subtraction. But step 3 uses the expanded form. This is a bit disjointed but not wrong. Let me check the severity definitions. 'Error' is a step or setup is wrong. 'Misleading' is a student would learn something false. 'Style' is correct but unclear. The jump from step 1 (an identity) to step 3 (an integral) skips stating the integral explicitly in a sentence, but step 3's comment says 'Area = ...'. This is acceptable. I will pass.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly identifies the intersection points, determines the correct upper and lower functions on the interval, and computes the definite integral accurately.
gpt-oss:20b: pass 2026-09-29
qwen3.6:27b-mlx: fail (error) 2026-09-29 — The integrand in step 1 is incorrect; the difference (top - bottom) is -3x^2 - 9x, but step 2 claims the top curve is -3x^2 - 6x - 1, which is false for x in (-3, 0). The line y = 3x - 1 is above the parabola in this interval, so the correct integrand should be (3x - 1) - (-3x^2 - 6x - 1) = 3x^2 + 9x, or the limits/signs must be handled to yield a positive area. The solution uses the negative of the correct difference, but since the integral of -3x^2 - 9x from -3 to 0 is positive (27/2), it accidentally gets the right magnitude but for the wrong reason (it integrates bottom - top, which is negative, but the bounds are reversed? No, bounds are -3 to 0. Let's check: integral of -3x^2 - 9x from -3 to 0. Antiderivative is -x^3 - 4.5x^2. At 0: 0. At -3: -(-27) - 4.5(9) = 27 - 40.5 = -13.5. Result 0 - (-13.5) = 13.5 = 27/2. So the calculation is correct for the integrand -3x^2 - 9x. However, is -3x^2 - 9x the correct top-bottom? Top is line 3x-1. Bottom is parabola -3x^2-6x-1. Top - Bottom = 3x - 1 - (-3x^2 - 6x - 1) = 3x^2 + 9x. The solution uses -3x^2 - 9x. This is -(Top - Bottom). Integrating -(Top - Bottom) from -3 to 0 gives -27/2. Wait. Let's re-evaluate. Integral of 3x^2 + 9x from -3 to 0. Antiderivative x^3 + 4.5x^2. At 0: 0. At -3: -27 + 40.5 = 13.5. Result 0 - 13.5 = -13.5. Area is absolute value, so 13.5. The solution integrates -3x^2 - 9x. This is negative of the correct difference. Integral is 13.5. So the number is right. But the sentence says 'y = -3x^2 - 6x - 1 is on top'. This is false. The line is on top. The setup is logically inconsistent and teaches false geometry.
gpt-oss:20b: pass 2026-09-29

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 2026-10-04 with SymPy 1.14.0.