Example 1 (Concluded)

Therefore |f "(x)| ≤ 1 for all x in [0,1]. We take M = 1 and use the error estimate given by the Trapezoidal Rule,

Thus our approximation is within an accuracy of 1/300,

This shows that the integral is, at least, between 1.146 and 1.154.

In this particular example we can even conclude that the integral is between 1.146 and 1.150 (rounded off to three places). That is, the integral is less than its trapezoidal approximation. This is because the second derivative f"(x) = (1 + x²)^-3/2 is always greater than 0, whence the curve is concave upwards and therefore y = f (x) is always less than or equal to the broken line used in the trapezoidal approximation. Actually, the value to three places is 1.148. This can be found by taking Δx = 1/10.