Answer: 1/2.
- A 1/2
- B 0
- C 1
- D 1/4
Correct answer: A. 1/2
Explanation: x<sup>3</sup>-x = x(x-1)(x+1), positive on (-1,0), negative on (0,1). Area = |integral -1 to 0| + |integral 0 to 1| = 1/4 + 1/4 = 1/2 (by symmetry of the odd function about the origin in absolute terms).
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.