Mathematics 2001 JAMB Past Questions - Question 39: Find the area bounded by the curves y=4-x² and y=2x+1

Correct Answer

B

Explanation

To find the area bounded by the curves y = 4 - x² and y = 2x + 1, we need to determine the points of intersection between the two curves.Setting the equations equal to each other, we have:4 - x² = 2x + 1Rearranging the equation, we get:x² + 2x - 3 = 0Factoring the quadratic equation, we have:(x + 3)(x - 1) = 0Setting each factor equal to zero, we find:x + 3 = 0 or x - 1 = 0Solving for x, we get:x = -3 or x = 1Now, we can find the y-values for these x-values by substituting them into either of the original equations.For x = -3:y = 4 - (-3)² = 4 - 9 = -5For x = 1:y = 4 - 1² = 4 - 1 = 3Therefore, the points of intersection are (-3, -5) and (1, 3).To find the area bounded by the curves, we need to integrate the difference between the two curves over the interval from x = -3 to x = 1.The area can be calculated as:A = ∫[from -3 to 1] (4 - x²) - (2x + 1) dxSimplifying, we have:A = ∫[from -3 to 1] (3 - x² - 2x) dxIntegrating, we get:A = [3x - (x³/3) - x²] [from -3 to 1]Evaluating the definite integral, we have:A = [(3(1) - (1³/3) - (1²)) - (3(-3) - ((-3)³/3) - ((-3)²))]A = [(3 - 1/3 - 1) - (-9 + 9/3 - 9)]A = [(3 - 1/3 - 1) - (-9 + 3 - 9)]A = [(3 - 1/3 - 1) - (-15)]A = [(3 - 1/3 - 1) + 15]A = [3 - 1/3 - 1 + 15]A = 17 - 1/3Therefore, the area bounded by the curves y = 4 - x² and y = 2x + 1 is 17 - 1/3 square units.