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JAMB 2012 Mathematics · Question 32

Evaluate ∫12(x2−4x)dx\int^2_1(x^2 - 4x)dx

Options

  1. A.113\frac{11}{3}
  2. B.311\frac{3}{11}
  3. C.−311-\frac{3}{11}
  4. D.−113-\frac{11}{3}

Answer: D. −113-\frac{11}{3}

Explanation written with AI and checked against the answer key

Analyze the question: We need the definite integral of x² − 4x from x = 1 to x = 2. Concept: For a definite integral, find the antiderivative F(x), then compute F(upper) − F(lower). Use the power rule: ∫xⁿ dx = xⁿ⁺¹/(n+1). Solution: Step 1: Integrate each term. ∫(x² − 4x) dx = x³/3 − 2x² Step 2: Evaluate at the upper limit (x = 2). F(2) = 8/3 − 2(4) = 8/3 − 8 = 8/3 − 24/3 = −16/3 Step 3: Evaluate at the lower limit (x = 1). F(1) = 1/3 − 2(1) = 1/3 − 6/3 = −5/3 Step 4: Subtract. F(2) − F(1) = −16/3 − (−5/3) = −16/3 + 5/3 = −11/3 Answer: −11/3, which is option D. Tip: Always write F(upper) − F(lower) with brackets so you do not lose a negative sign. Also, a quick sanity check: on [1, 2], x² − 4x is negative (e.g., at x = 1.5 it is 2.25 − 6 = −3.75), so the answer must be negative. That immediately eliminates A and B. Since the average value is about −3.7 over an interval of length 1, −11/3 ≈ −3.67 fits nicely.

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