Question

True or False For a given value of \(n,\) the Trapezoidal Rule with \(n\) subdivisions will always give a more accurate estimate of \(\int_{a}^{b} f(x) d x\) than a right Riemann sum with \(n\) subdivisions. Justify your answer.

Short answer

True. In general, for a given value of \(n,\) the Trapezoidal Rule with \(n\) subdivisions will provide a more accurate estimate of \(\int_{a}^{b} f(x) dx\) than a right Riemann sum with \(n\) subdivisions, especially if the function is not only increasing or decreasing.

Step-by-step solution

Understanding the difference between two methods

The first step is to understand the difference between the Trapezoidal rule and Right Riemann sum. The Trapezoidal Rule estimates the value of a definite integral by using trapezoids. It takes the average of the left-hand and right-hand sums to create a better approximation. In contrast, the Right Riemann sum uses rectangles to approximate the integral. It uses the right endpoint of each subinterval to determine the height of the rectangle. Hence, it might leave some parts uncovered or exceed the actual area under the graph when the function increses or decreases.

Assess Accuracy of Both Methods

The Trapezoidal Rule, given it uses the average of two points for estimation, generally provides a more accurate result compared to a Right Riemann sum. This is because it covers more area under the curve and can handle both increasing and decreasing functions better.

Provide Justification

While the Right Riemann sum might provide a more accurate result for an increasing function compared to the Left Riemann sum, it may not handle decreasing functions well. On the other hand, the Trapezoidal Rule, given its method of approximation, can manage both increasing and decreasing functions better and provide a more accurate approximation for general functions in most cases.