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Chapter 4: Problem 15

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.

Short Answer

Expert verified
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 by step solution

01

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.
02

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.
03

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.

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