Problem 17 - Calculus for Biology and Medicine

Question

How large should \(n\) be so that the midpoint rule approximation of $$ \int_{0}^{2} x^{2} d x $$ is accurate to within \(10^{-4} ?\) In Problems 18-24, use the theoretical error bound to determine how large \(n\) should be [Hint: In each case, find the second derivative of the integrand, graph it, and use a graphing calculator to find an upper bound on \(\left.\left|f^{\prime \prime}(x)\right| .\right]\)

Step-by-Step Solution

Answer

82 subintervals are needed for the approximation error to be within \(10^{-4}\).

Step 1: Understanding the Problem

We need to determine the number of subintervals, \(n\), required to approximate \(\int_{0}^{2} x^{2} \ dx\) using the midpoint rule, such that the approximation error is less than \(10^{-4}\). The midpoint rule error for \(n\) subintervals can be bounded by \(\frac{b-a}{24n^2}\cdot|f''(c)|\), where \(f''(c)\) is the maximum value of the absolute second derivative in the interval.

Step 2: Find the Second Derivative

The integrand is \(f(x) = x^2\). First, find the derivative: \(f'(x) = 2x\). Then, find the second derivative: \(f''(x) = 2\). Since \(f''(x)\) is constant, the maximum value of \(|f''(x)|\) on the interval \([0,2]\) is \(2\).

Step 3: Apply the Error Bound Formula

The error bound for the midpoint rule is given by \(\text{Error} \leq \frac{(b-a)^3}{24n^2}\cdot|f''(c)|\). Substituting \(a = 0\), \(b = 2\), and \(|f''(c)| = 2\), we have: \(\text{Error} \leq \frac{2^3}{24n^2} \cdot 2 = \frac{16}{24n^2} = \frac{2}{3n^2}\).

Step 4: Solve for n to Meet Error Requirement

We need \(\frac{2}{3n^2} \leq 10^{-4}\). Solving for \(n\), we have: \(3n^2 \geq 20000\), so \(n^2 \geq 6666.67\). Taking the square root, \(n \geq \sqrt{6666.67} \approx 81.65\). Since \(n\) must be an integer, round up to the nearest whole number, giving \(n = 82\).

Key Concepts

Midpoint Rule

The Midpoint Rule is a method used in numerical integration to approximate the value of a definite integral. It's particularly helpful when dealing with integrals that are difficult to evaluate analytically. Instead of calculating the area under a curve exactly, the Midpoint Rule uses rectangles, where the height of each rectangle is determined by the function value at the midpoint of each interval. This provides a simple way to estimate the total area.
When using the Midpoint Rule, keep in mind that the approximation improves as the number of subintervals, denoted as \(n\), increases. More subintervals mean that the rectangles better approximate the shape of the actual curve, leading to a more accurate result. For optimal results, carefully choose \(n\) based on the desired accuracy and complexity of the integrand.

Error Bound

In numerical integration, understanding the error bound is crucial for knowing how far off your approximation may be. When using the Midpoint Rule, the error can be bounded by a specific formula, often given by:

• \( \text{Error} \leq \frac{(b-a)^3}{24n^2} \cdot |f''(c)| \)

Here, \(a\) and \(b\) symbolize the interval limits, \(n\) is the number of subintervals, and \(|f''(c)|\) is the maximum absolute value of the second derivative over the interval.
This error bound gives a way to ensure that the approximation is within a tolerable margin, like \(10^{-4}\) in our given exercise. By adjusting \(n\), you can achieve a desired level of precision, which is key to the accuracy of numerical approximations.

Integral Approximation

Integral Approximation refers to the process of estimating the value of a definite integral. This is vital when the function is complex or not easily integrable analytically. Common techniques include the Midpoint Rule, Simpson's Rule, and the Trapezoidal Rule, among others. Each method has its own approach and level of accuracy.
The choice of method often depends on the function's behavior and the desired precision. In our exercise, we use the Midpoint Rule to approximate the integral \( \int_{0}^{2} x^{2} \ dx \). For this function, calculating the exact integral is straightforward, but practicing approximation techniques prepares you for more challenging integrals. Remember, a good approximation hinges on selecting an appropriate number of intervals and understanding the behavior of the function over the given domain.

Second Derivative

The Second Derivative of a function is a crucial concept in calculus, giving insight into the function's curvature or concavity. It is the derivative of the derivative, meaning it measures how the rate of change itself is changing. In the context of numerical integration, the second derivative is used to compute the error bound when approximating an integral.
For our specific function, \(f(x) = x^2\), the second derivative is constant: \(f''(x) = 2\). This constant second derivative simplifies the process of finding the error bound, as \(|f''(x)|\) is the same throughout the entire interval. When dealing with more complex functions, you might need to use calculus tools or graphing calculators to determine this maximum value, which is key to accurately bounding the numerical error.