Answer: 1/3.
- A 1/3
- B 2/3
- C 1
- D 1/6
Correct answer: A. 1/3
Explanation: By symmetry, consider x>=0: y=x and y=x<sup>2</sup> intersect at x=0,1. Area (x>=0 part) = integral 0 to 1 of (x-x<sup>2</sup>) dx = 1/6. Doubling for symmetry (x<=0 mirrors it) gives total area = 1/3.
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.