Chapter 6.5, Problem 32E

To determine

To find:The trapezoidal rule underestimate ${\displaystyle \underset{a}{\overset{b}{\int }}f(x)dx}$ is right Riemann sum is true or false.

Answer to Problem 32E

The trapezoidal rule underestimate ${\displaystyle \underset{a}{\overset{b}{\int }}f(x)dx}$ is right Riemann sum is false.

Explanation of Solution

Given information:

The trapezoidal rule underestimate ${\displaystyle \underset{a}{\overset{b}{\int }}f(x)dx}$ is right Riemann sum

Calculation:

Given statement is false.

Trapezoidal rule does not always give the better approximation than right Riemann sum approximation. Since, in trapezoidal rule we draw trapezoid at lower and upper point of interval and in Right rectangular method of approximation, we draw the rectangle at right-hand point of sub-intervals.

Take an example, if function is constant, either we draw rectangle of trapezoidal, it will be a rectangle. So, the approximation in the case of constant function is same for both.

Conclusion:

The approximation in the case of constant function is same for both.