MATHEMATICS · Single correct · A Level · Mathematics · Integration · Integration
Problem
A continuous function h intersects the x-axis at x=−2, x=1, and x=4. A student plans to compute the area of the region in the first quadrant bounded by h and the x-axis by evaluating ∫−24h(x)dx. Which of the following best explains why this choice of limits may yield an incorrect value for the desired area?
(a) Because h changes concavity on [−2,4], one integral cannot represent the area of regions with different curvature.
(b) Since the region lies entirely in the first quadrant, the lower limit must be x=0, not x=−2. A single integral from 0 to 4 would suffice.
(c) Portions of [−2,0] lie below the x-axis, so the integral over that interval would subtract, rather than add, area.
(d) The integral from −2 to 4 double-counts the area on [1,4], leading to an over-estimate.