Problem

How accurate is the Trapezoid Rule for approximating integrals? There is a theorem that tell us:

where T_n is the trapezoidal approximation of the integral on n subintervals of length h and M is a number such that

for all x in the interval [a, b].

Use this theorem as another way (besides calculating the exact value of the integral using integral formulas) to determine the accuracy of the trapezoid approximations in the first two examples.

Solution

In the second example, f(x) = x³ + x, so f ''(x) = 6x. In the interval [0, 9], the graph of f ''(x) is an upward-sloping straight line, so it reaches its maximum value at the right endpoint, 9.

so M = 54, h = 9/5, b = 9, a = 0. Then the error of our approximation,

is less than or equal to

This error bound 131.22 is greater than the actual error. Recall that in the second example we found the actual error using 5 trapezoids was about 65.61.

In the first example, the situation does not work out so well. We had

You can see the problem by looking at the graph of the second derivative.

In the interval [0, 2], |f ''(x)| does not have an upper bound, since it is not defined at zero, and the limit

approches –¥ as x approaches 0.

This shows that the theorem doesn't always gives us information about the error bound of a trapezpoid estimate to an integral, so it can't give us an indication of how accurate our approximation may be.