Answer: 32/3.
- A 32/3
- B 64/3
- C 16/3
- D 8
Correct answer: A. 32/3
Explanation: Intersections: x<sup>2</sup> = 4x gives x=0,4. Since 4x >= x<sup>2</sup> on [0,4], Area = integral from 0 to 4 of (4x-x<sup>2</sup>) dx = [2x<sup>2</sup> - x<sup>3</sup>/3] from 0 to 4 = 32 - 64/3 = 96/3-64/3 = 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.