Answer: 32/3.
- A 32/3
- B 16/3
- C 8
- D 0, since the curve is symmetric
Correct answer: A. 32/3
Explanation: y = x<sup>2</sup>-4 is below the x-axis on [-2,2] (since x<sup>2</sup><=4). Area = |integral from -2 to 2 of (x<sup>2</sup>-4) dx| = |[x<sup>3</sup>/3 - 4x] from -2 to 2| = |(8/3-8) - (-8/3+8)| = |-16/3 - 16/3| = 32/3.
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.