Problem 56AE

Exact Trapezoid Rule

Prove that the Trapezoid Rule is exact (no error) when approximating the definite integral of a linear function.

SOLUTION

Step 1

The trapezoidal method error formula is given by

for any in the interval .is the approximation of the definite integral using the trapezoidal rule .

A linear functions are those whose graph is a straight line.

A linear function is of the form

Thus the second derivative of a linear function is always zero.

Therefore for a linear function becomes zero.

Thus the Trapezoid Rule is exact (no error) when approximating the definite integral of a linear function.