If f is an increasing function on [a,b], then while if f is decreasing, then
From Figure N6–7 we infer that the area of a trapezoid is less than the true area if the graph of f is
concave down, but is more than the true area if the graph of f is concave up.
FIGURE N6–7
Figure N6–8 is helpful in showing how the area of a midpoint rectangle compares with that of a
trapezoid and with the true area. Our graph here is concave down. If M is the midpoint of AB, then the
midpoint rectangle is AM 1 M 2 B. We’ve drawn T 1 T 2 tangent to the curve at T (where the midpoint
ordinate intersects the curve). Since the shaded triangles have equal areas, we see that area AM 1 M 2
B = area AT 1 T 2 B.† But area AT 1 T 2 B clearly exceeds the true area, as does the area of the midpoint
rectangle. This fact justifies the right half of the inequality below; Figure N6–7 verifies the left half.
FIGURE N6–8
Generalizing to n subintervals, we conclude:
If the graph of f is concave down, then
If the graph of f is concave up, then
EXAMPLE 27
Write an inequality including L(n), R(n), M(n), T(n), and for the graph of f shown in Figure
N6–9.