JAMB 2023 Mathematics · Question 36
Determine the area of the region bounded by y = + 10 and Y = .
Options
- A.18
- B.
- C.
- D.
Answer: D.
Explanation written with AI and checked against the answer key
Step 1: Analyze the question We need the area enclosed between a parabola, y = 2x² + 10, and a line, y = 4x + 16. First we find where they meet, then integrate (top curve − bottom curve) between those points. Step 2: Find the intersection points Set the two equations equal: 2x² + 10 = 4x + 16 2x² − 4x − 6 = 0 x² − 2x − 3 = 0 (x − 3)(x + 1) = 0 So x = −1 and x = 3. Step 3: Decide which curve is on top Test x = 0: the line gives 16 and the parabola gives 10, so the line is above the parabola between the intersection points. Step 4: Set up the integral Area = ∫ from −1 to 3 of [(4x + 16) − (2x² + 10)] dx = ∫ from −1 to 3 of (−2x² + 4x + 6) dx Step 5: Integrate Antiderivative: F(x) = −(2/3)x³ + 2x² + 6x At x = 3: −(2/3)(27) + 2(9) + 18 = −18 + 18 + 18 = 18 At x = −1: −(2/3)(−1) + 2(1) − 6 = 2/3 + 2 − 6 = −10/3 Area = 18 − (−10/3) = 18 + 10/3 = 54/3 + 10/3 = 64/3 Step 6: Answer The area is 64/3 square units, which is option D. (Note: options A and B are partial values from the calculation, F(3) = 18 and F(−1) = −10/3. They are traps for students who stop before subtracting.) Quick tip: For the area between a parabola and a line, you can use the shortcut Area = |a|(x₂ − x₁)³/6. Here |a| = 2 and the difference is 4, so Area = 2 × 64 / 6 = 64/3. This is a fast way to check your answer in the exam hall.AI can make mistakes. Report anything that looks wrong.
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