Answer: 1/4.
- A 1/4
- B 1/2
- C 1/3
- D 1/6
Correct answer: A. 1/4
Explanation: On [0,1], x >= x<sup>3</sup>, so Area = integral from 0 to 1 of (x - x<sup>3</sup>) dx = [x<sup>2</sup>/2 - x<sup>4</sup>/4] from 0 to 1 = 1/2 - 1/4 = 1/4.
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.