Answer: 32/3.
- A 32/3
- B 16/3
- C 8/3
- D 64/3
Correct answer: A. 32/3
Explanation: Upper half: y = 2sqrt(x). Area (upper half only) = integral from 0 to 4 of 2sqrt(x) dx = 2 times [(2/3)x<sup>3/2</sup>] from 0 to 4 = 2 times (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.