Answer: 9/2.
- A 9/2
- B 9
- C 3
- D 27/6 is the unsimplified value, not the final answer
Correct answer: A. 9/2
Explanation: Intersections: x<sup>2</sup> = x+2 gives x=-1,2. Area = integral from -1 to 2 of [(x+2)-x<sup>2</sup>] dx = [x<sup>2</sup>/2+2x-x<sup>3</sup>/3] from -1 to 2 = (2+4-8/3)-(0.5-2+1/3) = (10/3) - (-7/6) = 27/6 = 9/2.
The definite integral ∫ₐᵇf(x)dx computes the exact area of the shaded region bounded by the curve, the x-axis, and the vertical lines x=a and x=b - the same idea behind the Riemann sum, but evaluated exactly rather than approximated by rectangles.
Concept context
Using definite integrals to compute the area under a curve, the area between two curves, and the areas enclosed by standard curves like circles, parabolas, and ellipses.