Question

Estimate the minimum number of subintervals needed to approximate the integral \(\int_{2}^{3}\left(2 x^{3}+4 x\right) d x\) with an error of magnitude less than \(0.0001\) using the trapezoidal rule.

Short answer

The minimum number of subintervals needed is 174.

Step-by-step solution

Understand the Trapezoidal Rule Formula

The trapezoidal rule is used to approximate the integral of a function. The formula is given by: \[ T_n = \frac{b-a}{2n} \left(f(a) + 2 \sum_{i=1}^{n-1} f(x_i) + f(b) \right) \]where \( n \) is the number of subintervals, and \( a \) and \( b \) are the limits of integration.

Determine the Error Bound Formula

The error \( E \) for the trapezoidal rule can be estimated using the formula:\[ E \leq \frac{(b-a)^3}{12n^2} \max_{x \in [a, b]} |f''(x)| \]We need to find \( n \) such that the error \( E < 0.0001 \).

Find the Second Derivative of the Function

The given function is \( f(x) = 2x^3 + 4x \). First, find its second derivative:\[ f'(x) = 6x^2 + 4 \]\[ f''(x) = 12x \]

Evaluate the Maximum of the Second Derivative

On the interval \([2,3]\), evaluate \( f''(x) = 12x \) to find its maximum:This gives \( f''(2) = 24 \) and \( f''(3) = 36 \).Thus, \( \max_{x \in [2, 3]} |f''(x)| = 36 \).

Use Error Bound to Find Required n

Use the error bound formula:\[ \frac{(3-2)^3}{12n^2} \times 36 < 0.0001 \]Simplifying gives:\[ \frac{36}{12n^2} < 0.0001 \]\[ \frac{3}{n^2} < 0.0001 \]\[ n^2 > \frac{3}{0.0001} = 30000 \]\[ n > \sqrt{30000} \approx 173.21 \]So, \( n \) must be at least 174.

Key concepts

Error Estimation in Trapezoidal Rule

The Trapezoidal Rule is a numerical method used to approximate definite integrals. However, like all approximation methods, it introduces an error. Understanding this error is vital. The error estimation formula for the Trapezoidal Rule helps us determine how closely our approximation matches the actual integral.

The error formula for the Trapezoidal Rule is:

• \(E \leq \frac{(b-a)^3}{12n^2} \max_{x \in [a, b]} |f''(x)| \)

This formula indicates that the error depends on the width of the interval \((b-a)\), the square of the number of subintervals \((n^2)\), and the maximum value of the second derivative of the function \(f''(x)\) within the interval. To reduce the error, you can either increase the number of subintervals \(n\) or find a function with a smaller maximum second derivative over the interval.

In our example, to ensure the error remains below 0.0001, we must solve for \(n\) in the context of the given error bound. Calculating this helps us determine the minimum number of subintervals needed to achieve the desired accuracy.

Calculus Behind the Trapezoidal Rule

Calculus plays a crucial role in understanding and applying the Trapezoidal Rule. At its core, the Trapezoidal Rule uses a linear approximation of the curve to estimate the area under a function. This is directly linked to the integral calculus, where finding the area under the curve is a central problem.

In the context of the original problem, we start with differentiating the given function to find its second derivative \(f''(x)\). We deduced \(f(x) = 2x^3 + 4x\) to:

• First derivative: \(f'(x) = 6x^2 + 4\)
• Second derivative: \(f''(x) = 12x\)

The importance of the second derivative lies in its ability to measure the curvature of \(f(x)\) over the interval \([2,3]\). The more the curve bends, the more subintervals are needed to accurately approximate the area under the curve using straight lines (trapezoids). Understanding how the derivatives reflect changes in the function's slope prepares us to evaluate and manage the error.

Integral Approximation Using the Trapezoidal Rule

Integral approximation becomes often necessary when a function cannot be easily integrated analytically, or when the function is only known from measurements. The Trapezoidal Rule is an accessible method for this approximation. It divides the interval into smaller subintervals and approximates the area under the curve as a series of trapezoids.

To implement this for the integral \(\int_{2}^{3}(2x^3 + 4x)dx\), we address:

• The interval \( [2, 3] \) into \( n \) equal subintervals
• Apply the Trapezoidal Rule formula
• Adjust \( n \) such that the error criterion \( E < 0.0001 \) is satisfied

Practically, this requires solving for the minimum \( n \) ensuring our approximation meets the precision standards. By analyzing the problem computationally, as detailed in the given solution steps, we confirm that at least 174 subintervals will achieve this. Utilizing software or a calculator facilitates this computation when dealing with complex or lengthy expressions.