Question

Suppose \(f\) is positive and its first two derivatives are continuous on \([a, b] .\) If \(f^{\prime \prime}\) is positive on \([a, b],\) then is a Trapezoid Rule estimate of \(\int_{a}^{b} f(x) d x\) an underestimate or overestimate of the integral? Justify your answer using Theorem 2 and an illustration.

Short answer

Based on the given conditions and Theorem 2, the trapezoid rule estimate of the integral \(\int_{a}^b f(x) dx\) is an underestimate. This is because the function \(f(x)\) is concave up on the interval \([a, b]\), and when a function is concave up, the trapezoid rule yields an underestimate of the integral. This can also be visually confirmed by noting that the trapezoids lie below the curve, resulting in their combined area being smaller than the actual integral value.

Step-by-step solution

Identify the given information

We are given that: 1. \(f\) is a positive function. 2. The first two derivatives of \(f\) are continuous on \([a, b]\). 3. \(f^{\prime \prime} > 0\) on \([a, b]\). We need to determine if the trapezoid rule estimate of \(\int_{a}^{b} f(x) dx\) is an overestimate or an underestimate of the integral using Theorem 2 and an illustration.

Recall Theorem 2 and trapezoid rule

Theorem 2 states that when a function is concave up, the trapezoid rule will yield an underestimate of the integral, and when a function is concave down, the trapezoid rule will yield an overestimate. Trapezoid rule is an approximation method for calculating \(\int_{a}^b f(x) dx\). It estimates the integral by approximating the region under the curve of \(f(x)\) using trapezoids and summing up their areas.

Determine if the function is concave up or down

Since \(f^{\prime \prime}(x) > 0\) on \([a, b]\), it means that the function is concave up on the given interval (since the second derivative is positive).

Apply Theorem 2 to find the nature of the trapezoid rule estimate

As the function is concave up, Theorem 2 tells us that the trapezoid rule will provide an underestimate of the integral \(\int_{a}^b f(x) dx\).

Illustration

Since the function is concave up, imagine a curve that is shaped like a "smile." When using the trapezoid rule, the trapezoids will be drawn below the curve, and hence their combined area will be smaller than the actual integral value. This illustration confirms our answer that the trapezoid rule estimate is indeed an underestimate of the integral.

Key concepts

Trapezoid Rule

The Trapezoid Rule is a straightforward method for approximating definite integrals. It's like making trapezoids under the curve of a function and adding their areas to estimate the total area under the curve. The trapezoid, a simple geometric shape, has straight edges on top and bottom, but when used to approximate a curve, these straight edges touch the curve at only two points - right at the endpoints of an interval.
To use the Trapezoid Rule for an integral \([a, b]\), we

• divide the interval into smaller sub-intervals,
• form a trapezoid for each sub-interval using the function values at the endpoints,
• and then sum up the areas of all these trapezoids to get the estimate.

The mathematical expression for the Trapezoid Rule is given by:\[\int_{a}^{b} f(x) \, dx \approx \frac{b-a}{2} [f(a) + f(b)]\]The approximation becomes more accurate as the number of trapezoids increases, but this depends on the behavior of the function itself.

Concavity

Concavity refers to the curvature of a function. If a function is "concave up," it looks like a bowl that holds water. Conversely, a function that is "concave down" appears as a dome. The second derivative of a function, \(f''(x)\), is used to determine concavity:

• If \(f''(x) > 0\), the function is concave up.
• If \(f''(x) < 0\), the function is concave down.

Concavity has an important role in integral estimations using methods like the Trapezoid Rule. When a function is concave up, as it is in this exercise where \(f''(x) > 0\) for all \(x\) in \([a, b]\), the curve always lies above the tops of the trapezoids. This means the Trapezoid Rule will "catch" less area than is actually there, leading to an underestimate of the definite integral.

Integral Estimation

Integral estimation is a useful technique when an exact evaluation of integrals is not feasible. This approximation can be conducted by employing various techniques, like the Trapezoid Rule, Riemann sums, or Simpson's Rule. Each method has its particular efficiency and accuracy points. When it comes to functions that aren't easily integrable using standard calculus techniques, or when a quick approximation is needed, methods like the Trapezoid Rule step in. The key goal here is to get a practical close estimate of the total area under the curve from \(a\) to \(b\).
Using the Trapezoid Rule becomes particularly insightful when understanding how concavity affects the estimates:

• Concave up yields underestimates with the Trapezoid Rule.
• Concave down results in overestimates with the same method.

Integral estimation methods like this allow for practical applications in various fields, such as physics and engineering, where quick and efficient area approximations are essential.