Answer: 32/3.
- A 16/3
- B 32/3
- C 8/3
- D 8
Correct answer: B. 32/3
Explanation: Intersections at x = -2, 2. Area = integral from -2 to 2 of (4 - x<sup>2</sup>) dx = 2 times integral from 0 to 2 of (4-x<sup>2</sup>) dx = 2[4x - x<sup>3</sup>/3] from 0 to 2 = 2(8 - 8/3) = 2(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.