Problem 15 - Essential Calculus

Question

If the function \(f\) is concave upward on the interval \([a, b]\), will the Trapezoidal Rule yield a result greater than or less than \(\int_{a}^{b} f(x) d x ?\) Explain.

Step-by-Step Solution

Answer

When \(f(x)\) is concave upward on the interval \([a, b]\), the Trapezoidal Rule will yield a result that is less than \(\int_{a}^{b} f(x) dx\).

Step 1: Understanding the Trapezoidal Rule

The Trapezoidal Rule is a numerical approximation method for definite integrals. It works by approximating the area under the curve of the function by a series of trapezoids. For an integral over the interval \([a, b]\), the base of each trapezoid spans a small segment of the interval, and the height of the trapezoid is based on the function's values at the ends of the interval.

Step 2: Understanding the Concavity

If a function \(f(x)\) is concave upward on \([a, b]\), this means that the curve of \(f(x)\) in that interval lies entirely above any line segment connecting any two points on the curve. This implies that if we draw a trapezoid under the curve, there will be additional area under the curve above the trapezoid.

Step 3: Comparing the Trapezoidal Rule and the Definite Integral

Considering the previous two steps, we can conclude that the Trapezoid rule will underestimate the area under the curve of a function that is concave upward. Since \(\int_{a}^{b} f(x) dx\) computes the exact area under the curve of \(f(x)\) between \(a\) and \(b\), the Trapezoidal Rule approximation will be less than \(\int_{a}^{b} f(x) dx\) for a function that is concave upward.