Problem 20 - Thomas Calculus

Question

In Exercises \(15-26,\) estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than \(10^{-4}\) by ( a ) the Trapezoidal Rule and (b) Simpson's Rule. (The integrals in Exercises \(15-22\) are the integrals from Exercises \(1-8 .\) ) $$ \int_{-1}^{1}\left(t^{3}+1\right) d t $$

Step-by-Step Solution

Answer

Trapezoidal Rule: 82 subintervals; Simpson's Rule: 1 subinterval.

Step 1: Identify Basic Information

We have the integral \( \int_{-1}^{1}(t^3 + 1)\, dt \) and we are asked to approximate this integral with a specified error using the Trapezoidal Rule and Simpson's Rule. We need to find the minimum number of subintervals \( n \) for which the corresponding error is less than \( 10^{-4} \).

Step 2: Trapezoidal Rule Error Formula

The error bound for the Trapezoidal Rule is given by:\[ E_T = \frac{(b-a)^3}{12n^2} M_2 \]where \( M_2 \) is the maximum absolute value of the second derivative of the function within the interval \([a, b]\). Here, \( a = -1 \), \( b = 1 \).

Step 3: Calculate the Second Derivative

The given function is \( t^3 + 1 \). Its first derivative is \( 3t^2 \) and the second derivative is \( 6t \). The maximum absolute value of \( 6t \) on \([-1, 1]\) is 6, which occurs at \( t = \pm 1 \). Therefore, \( M_2 = 6 \).

Step 4: Set Up the Trapezoidal Rule Error Inequality

We must have:\[ \frac{(1 - (-1))^3}{12n^2} \times 6 < 10^{-4} \]This simplifies to:\[ \frac{8}{12n^2} < 10^{-4} \]

Step 5: Solve for Minimum n (Trapezoidal Rule)

Continuing from the inequality:\[ \frac{2}{3n^2} < 10^{-4} \]\[ n^2 > \frac{2}{3 \times 10^{-4}} \]\[ n^2 > 6666.67 \]Taking the square root gives \( n > 81.65 \). Thus, the minimum \( n \) must be 82.

Step 6: Simpson's Rule Error Formula

The error bound for Simpson's Rule is:\[ E_S = \frac{(b-a)^5}{180n^4} M_4 \]where \( M_4 \) is the maximum absolute value of the fourth derivative of the function within \([a, b]\).

Step 7: Calculate the Fourth Derivative

The given function \( t^3 + 1 \) leads to the first, second, third, and fourth derivatives as \( 3t^2 \), \( 6t \), \( 6 \), and \( 0 \) respectively. Thus, \( M_4 = 0 \) because the fourth derivative is constant.

Step 8: Set the Condition for Simpson's Rule

Since \( M_4 = 0 \), the error \( E_S \) is zero. Therefore, Simpson's Rule can satisfy the error requirement with any positive number of intervals, including \( n = 1 \).

Step 9: Conclusion

For the Trapezoidal Rule, we require at least 82 subintervals to ensure the error is less than \( 10^{-4} \). For the Simpson's Rule, since the error is zero for any number of subintervals, \( n = 1 \) is sufficient to achieve the required accuracy.

Key Concepts

Trapezoidal Rule

The Trapezoidal Rule is a straightforward numerical method for approximating the definite integral of a function. It works by dividing the overall interval \([a, b]\) into smaller subintervals of equal width. The function is then approximated by a series of trapezoids, whose areas are summed to provide the integral's approximation. This method is particularly simple as it relies on linear approximations.

The error estimate of the Trapezoidal Rule is crucial to determine how many subintervals are necessary for a specific accuracy. The error formula used is \[ E_T = \frac{(b-a)^3}{12n^2} M_2 \] where \(a\) and \(b\) are the integration limits, \(n\) is the number of subintervals, and \(M_2\) is the maximum absolute value of the second derivative on \([a, b]\). This error formula shows that the approximation gets more accurate as the number of subintervals increases.

For the function \(t^3 + 1\) integrated from -1 to 1, we found \(M_2 = 6\) by calculating that the maximum of the absolute value of the second derivative \(6t\) occurs at \(t = -1\) or \(t = 1\). Setting up the inequality \[ \frac{8}{12n^2} < 10^{-4} \] and solving it indicates that \(n\) must be at least 82 to ensure the error is below \(10^{-4}\).

Simpson's Rule

Simpson's Rule is another method for numerical integration that is often more accurate than the Trapezoidal Rule. It uses parabolic segments instead of straight lines to approximate the curve of the function over each subinterval. This typically provides a better estimate, especially for smooth functions.

Simpson’s Rule requires that the number of subintervals \(n\) is even, and its error is given by \[ E_S = \frac{(b-a)^5}{180n^4} M_4 \] where \(M_4\) is the maximum absolute value of the fourth derivative of the function on \([a, b]\). For the integral of \(t^3 + 1\) from -1 to 1, the fourth derivative is zero because the third derivative \(6\) is a constant, resulting in \(M_4 = 0\).

Given that \(M_4 = 0\), it turns out the error is zero no matter the choice of \(n\). This interesting result implies Simpson's Rule already satisfies the error requirement without needing many subintervals. In fact, just \(n = 1\) is sufficient to achieve an error of magnitude less than \(10^{-4}\) in this case.

Error Estimation

Error estimation in numerical integration indicates how close an approximation is to the actual value of an integral. Having precise control over this error is essential for applications requiring a high level of accuracy, often encountered in engineering and scientific computations.

For methods like the Trapezoidal and Simpson's Rules, the error is influenced by the number of subintervals used and the behavior of the function's derivatives. In general:

• The Trapezoidal Rule error decreases proportionally with the square of the number of subintervals \(n^2\).
• The Simpson's Rule error decreases even more rapidly, with the fourth power of \(n\).

The crucial idea here is that more subintervals lead to smaller errors, but the relationship differs between the two methods. The Trapezoidal Rule is simpler but demands more subintervals to reach similar accuracies compared to Simpson’s Rule.

Understanding these error estimations facilitates deciding on the right numerical method and how many computations are necessary to achieve your desired precision level.