Answer: 27/3 = 9.
- A 9
- B 27/3 = 9
- C 8
- D 10
Correct answer: B. 27/3 = 9
Explanation: Area = integral from -1 to 2 of (4-x<sup>2</sup>) dx = [4x - x<sup>3</sup>/3] from -1 to 2 = (8-8/3) - (-4+1/3) = (16/3) - (-11/3) = 27/3 = 9.
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.