Answer: 2.5.
- A 2
- B 2.5
- C 3
- D 1.5
Correct answer: B. 2.5
Explanation: Split at x=1: integral from 0 to 1 of (1-x) dx = [x - x<sup>2</sup>/2] from 0 to 1 = 1 - 0.5 = 0.5. Integral from 1 to 3 of (x-1) dx = [x<sup>2</sup>/2 - x] from 1 to 3 = (4.5-3) - (0.5-1) = 1.5 + 0.5 = 2. Total area = 0.5 + 2 = 2.5.
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.