91Ó°ÊÓ

The Trapezoidal Rule is a straightforward numerical method to estimate the value of a definite integral. It simplifies the area under a curve by breaking it down into a series of trapezoids, which can be easily calculated and summed up. In this exercise, the integral \( \int_{0}^{2} (t^3 + t) \, dt \) is approximated using the Trapezoidal Rule with \( n=4 \) steps.
First, the interval \([0, 2]\) is divided into 4 subintervals, each of width \( h = 0.5 \). Function values are evaluated at these points: \( f(0), f(0.5), f(1), f(1.5) \), and \( f(2) \). The formula applied is:

• \( T_4 = \frac{0.5}{2} [f(0) + 2f(0.5) + 2f(1) + 2f(1.5) + f(2)] \)

Once the values are substituted and calculated, the summed area approximates the area under the curve.In addition, the upper bound error of the Trapezoidal Rule \( |E_T| \) depends on the second derivative of the function. For \( f(t) = t^3 + t \), the second derivative is \( f''(t) = 6t \). The maximum value of this function over the given interval helps define the potential error range.

Simpson’s Rule offers a more accurate method for numerical integration, especially when the function outline is parabolic between points. Like the Trapezoidal Rule, Simpson’s Rule requires dividing the interval into even segments, but it applies a quadratic polynomial to approximate the function.For the integral \( \int_{0}^{2} (t^3 + t) \, dt \), with \( n=4 \), Simpson's Rule uses the same points as the Trapezoidal Rule but with a different formula:

• \( S_4 = \frac{0.5}{3} [f(0) + 4f(0.5) + 2f(1) + 4f(1.5) + f(2)] \)

This formula involves alternating coefficients of 4 and 2 to allow for the quadratic interpolation.Simpson's Rule proves to be highly precise because the error bound depends on the fourth derivative of the function. For this specific function, \( f^{(4)}(t) = 0 \), leading to a calculated error bound of zero, demonstrating the method's effectiveness in certain scenarios.

Error analysis in numerical integration helps understand the accuracy of our approximations and provides insights into improving these methods. For the Trapezoidal Rule, the error analysis uses the formula:

• \( |E_T| \leq \frac{(b-a)^3}{12n^2} \max_{a \leq t \leq b} |f''(t)| \)

Here, a higher value of \( n \) (more subintervals) or a function with a smaller second derivative curve decreases the error. Calculating \( |E_T| \) directly involves comparing the approximated integral with the true integral.Error for Simpson's Rule is traditionally smaller due to its quadratic approximation technique. The error bound relies on the fourth derivative:

• \( |E_S| \leq \frac{(b-a)^5}{180n^4} \max_{a \leq t \leq b} |f^{(4)}(t)| \)

In cases such as with \( t^3 + t \), where \( f^{(4)}(t) = 0 \), the theoretical error is non-existent, guaranteeing high precision.Expressing errors as percentages of the true integral values aids in grasping their significance. However, a value close to zero reflects excellent accuracy, making Simpson's Rule particularly valuable for precise calculations.