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Area between curves

Problem 5.94 · medium Video

Find the area of the region bounded by \( \displaystyle y = - 2 x^{2} + 13 x - 23 \) and \( \displaystyle y = 1 - x \).

\[ \left(8 - 2 x\right) \left(x - 3\right) = - 2 x^{2} + 14 x - 24 \]
Set the curves equal: top − bottom, factored.✓ Proved
They cross at x = 3 and x = 4; between them y = -2*x**2 + 13*x - 23 is on top.
\[ \int\limits_{3}^{4} \left(- 2 x^{2} + 14 x - 24\right)\, dx = \frac{1}{3} \]
Area = ∫ (top − bottom) dx.✓ Proved

Answer \( \frac{1}{3} \)

The curves y = -2x² + 13x - 23 and y = 1 - x between x = 3 and x = 4. Thin rectangles fill the gap between them, getting narrower until they fill the region exactly; its area is 1/3.

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 setup in step 1 incorrectly factors the difference of the functions. The actual difference is -2x^2 + 14x - 24, but the problem states the parabola is -2x^2 + 13x - 23. The difference should be (-2x^2 + 13x - 23) - (1 - x) = -2x^2 + 14x - 24. Wait, let me re-calculate. (-2x^2 + 13x - 23) - (1 - x) = -2x^2 + 13x + x - 23 - 1 = -2x^2 + 14x - 24. The factorization -2(x-3)(x-4) = -2(x^2 - 7x + 12) = -2x^2 + 14x - 24. So the algebra in step 1 is actually correct for the difference. However, step 2 claims the curves cross at x=3 and x=4. Let's check the original equations. -2x^2 + 13x - 23 = 1 - x => -2x^2 + 14x - 24 = 0 => x^2 - 7x + 12 = 0 => (x-3)(x-4)=0. So the intersection points are correct. The integral calculation is correct. Why did I think it was wrong? Let me re-read carefully. Problem: y = -2x^2 + 13x - 23 and y = 1 - x. Difference: (-2x^2 + 13x - 23) - (1 - x) = -2x^2 + 14x - 24. Roots of -2x^2 + 14x - 24 = 0 are x=3, x=4. Integral from 3 to 4 of (-2x^2 + 14x - 24) dx. Antiderivative: -2/3 x^3 + 7x^2 - 24x. Evaluated at 4: -2/3(64) + 7(16) - 24(4) = -128/3 + 112 - 96 = -128/3 + 16 = (-128 + 48)/3 = -80/3. Evaluated at 3: -2/3(27) + 7(9) - 24(3) = -18 + 63 - 72 = -27. Difference: -80/3 - (-27) = -80/3 + 81/3 = 1/3. The solution is actually correct. The note in step 1 says "Set the curves equal: top - bottom, factored". The equation shown is `-2*(x - 4)*(x - 3), -2*x**2 + 14*x - 24`. This represents the integrand. The sentence in step 2 correctly identifies the bounds and which function is on top (since the coefficient of x^2 is negative, the parabola opens downward, so it is above the line between the roots). The integral result is correct. Wait, looking closer at Step 1's label: "Set the curves equal: top - bottom, factored". The equation line shows the factored form and the expanded form of the *difference*. It doesn't show the equation set to 0, but it shows the integrand. This is a bit loose in description but mathematically sound for the purpose of integration. Is there any error? Let's check the function values. At x=3.5: Parabola: -2(3.5)^2 + 13(3.5) - 23 = -2(12.25) + 45.5 - 23 = -24.5 + 45.5 - 23 = 21 - 23 = -2. Line: 1 - 3.5 = -2.5. Parabola (-2) > Line (-2.5). So Parabola is on top. Correct. The solution seems perfectly correct. Why would it fail? Maybe the prompt implies I should find an error if one exists, but if none exists, I pass. Let me double check the problem statement vs the solution text. Problem: y = -2x^2 + 13x - 23. Solution Step 1: -2*x**2 + 14*x - 24. This is the difference. Step 2: Cross at 3 and 4. Correct. Step 3: Integral is 1/3. Correct. It seems correct. I will pass it.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-09-27 — The setup in step 1 incorrectly factors the difference of the functions. The actual difference is -2x^2 + 14x - 24, but the problem states the parabola is -2x^2 + 13x - 23. The difference should be (-2x^2 + 13x - 23) - (1 - x) = -2x^2 + 14x - 24. Wait, let me re-calculate. (-2x^2 + 13x - 23) - (1 - x) = -2x^2 + 13x + x - 23 - 1 = -2x^2 + 14x - 24. The factorization -2(x-3)(x-4) = -2(x^2 - 7x + 12) = -2x^2 + 14x - 24. So the algebra in step 1 is actually correct for the difference. However, step 2 claims the curves cross at x=3 and x=4. Let's check the original equations. -2x^2 + 13x - 23 = 1 - x => -2x^2 + 14x - 24 = 0 => x^2 - 7x + 12 = 0 => (x-3)(x-4)=0. So the intersection points are correct. The integral calculation is correct. Why did I think it was wrong? Let me re-read carefully. Problem: y = -2x^2 + 13x - 23 and y = 1 - x. Difference: (-2x^2 + 13x - 23) - (1 - x) = -2x^2 + 14x - 24. Roots of -2x^2 + 14x - 24 = 0 are x=3, x=4. Integral from 3 to 4 of (-2x^2 + 14x - 24) dx. Antiderivative: -2/3 x^3 + 7x^2 - 24x. Evaluated at 4: -2/3(64) + 7(16) - 24(4) = -128/3 + 112 - 96 = -128/3 + 16 = (-128 + 48)/3 = -80/3. Evaluated at 3: -2/3(27) + 7(9) - 24(3) = -18 + 63 - 72 = -27. Difference: -80/3 - (-27) = -80/3 + 81/3 = 1/3. The solution is actually correct. The note in step 1 says "Set the curves equal: top - bottom, factored". The equation shown is `-2*(x - 4)*(x - 3), -2*x**2 + 14*x - 24`. This represents the integrand. The sentence in step 2 correctly identifies the bounds and which function is on top (since the coefficient of x^2 is negative, the parabola opens downward, so it is above the line between the roots). The integral result is correct. Wait, looking closer at Step 1's label: "Set the curves equal: top - bottom, factored". The equation line shows the factored form and the expanded form of the *difference*. It doesn't show the equation set to 0, but it shows the integrand. This is a bit loose in description but mathematically sound for the purpose of integration. Is there any error? Let's check the function values. At x=3.5: Parabola: -2(3.5)^2 + 13(3.5) - 23 = -2(12.25) + 45.5 - 23 = -24.5 + 45.5 - 23 = 21 - 23 = -2. Line: 1 - 3.5 = -2.5. Parabola (-2) > Line (-2.5). So Parabola is on top. Correct. The solution seems perfectly correct. Why would it fail? Maybe the prompt implies I should find an error if one exists, but if none exists, I pass. Let me double check the problem statement vs the solution text. Problem: y = -2x^2 + 13x - 23. Solution Step 1: -2*x**2 + 14*x - 24. This is the difference. Step 2: Cross at 3 and 4. Correct. Step 3: Integral is 1/3. Correct. It seems correct. I will pass it.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The setup in line 1 incorrectly factors the difference of the functions. The actual difference is -2x^2 + 14x - 24, but the problem states the parabola is -2x^2 + 13x - 23. The difference should be (-2x^2 + 13x - 23) - (1 - x) = -2x^2 + 14x - 24. Wait, let me re-calculate. (-2x^2 + 13x - 23) - (1 - x) = -2x^2 + 14x - 24. The roots of -2x^2 + 14x - 24 = 0 are x=3 and x=4. So the algebra in line 1 is actually correct for the difference. However, line 2 claims the curves cross at 3 and 4, which is correct. But let's check the area calculation. Integral from 3 to 4 of (-2x^2 + 14x - 24) dx. Antiderivative is -2/3 x^3 + 7x^2 - 24x. Evaluated at 4: -128/3 + 112 - 96 = -128/3 + 16 = -80/3. Evaluated at 3: -54/3 + 63 - 72 = -18 - 9 = -27 = -81/3. Difference: -80/3 - (-81/3) = 1/3. The calculation is correct. Why did I think it was wrong? Let me re-read the problem statement carefully. y = -2x^2 + 13x - 23 and y = 1 - x. Difference: -2x^2 + 14x - 24. Roots: 2x^2 - 14x + 24 = 0 => x^2 - 7x + 12 = 0 => (x-3)(x-4)=0. Roots 3, 4. Integral is 1/3. The solution seems correct. Is there a subtle error? Line 1 says '-2*(x - 4)*(x - 3), -2*x**2 + 14*x - 24'. This is the factored form and expanded form of the difference. Line 2 says 'They cross at x = 3 and x = 4; between them y = -2*x**2 + 13*x - 23 is on top.' This is correct because the parabola opens downward and the line is below it between the roots. Line 3 calculates the integral. The result is 1/3. The solution appears sound. Wait, looking closer at the prompt's instruction: 'Each line of the solution is either an EQUATION... or a SENTENCE'. Line 1 is marked [equation, proved]. It shows the factored form and the expanded form. It doesn't explicitly state 'Difference = ...'. But the comment says 'Set the curves equal: top - bottom, factored.' This is slightly ambiguous phrasing ('Set the curves equal' usually implies finding intersection, but the content is the difference). However, the math is correct. Is there a 'misleading' aspect? The phrase 'Set the curves equal' is technically incorrect terminology for forming the integrand (which is 'subtract the curves'), but the result is correct. A student might be confused by 'set equal' vs 'subtract'. But is it an error? 'Set the curves equal' is the method to find bounds. The line content is the difference. The comment is slightly imprecise. Let's look for a harder error. Did I misread the problem? y = -2x^2 + 13x - 23. y = 1 - x. Difference: -2x^2 + 14x - 24. Correct. Bounds 3, 4. Correct. Integral 1/3. Correct. The only potential issue is the comment in line 1 'Set the curves equal'. This is a description of the *intent* to find bounds, but the line *content* is the difference. This is a style/clarity issue, not a mathematical error. However, often these adversarial reviews look for subtle mistakes. Let me double check the arithmetic one more time. F(x) = -2/3 x^3 + 7x^2 - 24x. F(4) = -128/3 + 112 - 96 = -128/3 + 16 = (-128+48)/3 = -80/3. F(3) = -54/3 + 63 - 72 = -18 - 9 = -27 = -81/3. F(4)-F(3) = 1/3. The math is perfect. The text 'Set the curves equal' is just a bad comment for a line that computes the difference. It might be 'misleading' if a student thinks you set them equal to get the integrand. But the line is marked 'proved' by a CAS, so the algebra is trusted. The sentence in line 2 is correct. The sentence in line 3 is correct. I will mark it as pass because the math is correct and the 'error' is just a slightly sloppy comment in a line that is otherwise algebraically verified. Actually, wait. If I have to be adversarial, is 'Set the curves equal' wrong enough to fail? It describes the process of finding the limits, which are used in the integral. It's not describing the integrand itself. It's acceptable shorthand. I'll pass it.
gpt-oss:20b: pass 2026-09-26
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The setup in line 1 computes the difference for y = -2x^2 + 14x - 24, but the problem specifies y = -2x^2 + 13x - 23. The coefficients and intersection points are incorrect for the given problem.
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 2026-09-27 with SymPy 1.14.0.