Use inscribed rectangles to approximate the area under f(x) = –x2 + 7 for the –2 ≤ x ≤ 0 and rectangle width 0.5.

Let us find how many rectangles we needsince rectangle width, δx, for approximating the area under a curve on the interval a ≤ x  ≤ b  with n rectangles is    we solve for n to get    we need 4 rectangles.since f(x) = -x² + 7 is a parabola with a vertex at (0,7) and it opens downwards, it is increasing on the interval  -2 ≤ x ≤ 0 (as it approaches the vertex from the left, it reaches a maximum point).to use inscribed rectangles means to have an area that is an underestimate. to underestimate the area of an increasing function, we use left endpoints    the approximate area is 10.25