91Ó°ÊÓ

The Midpoint Approximation is a numerical technique used to estimate the value of a definite integral. It works by dividing the integral's interval into smaller, equally spaced subintervals. Instead of using the endpoints of these subintervals, we evaluate the function at the midpoint of each subinterval. This provides a more central estimate of the area under the curve.

In this exercise, we approximate the integral \( \int_{1}^{3} e^{-2x} \, dx \) using the midpoint approximation with 10 subintervals. Here's the breakdown:

• Calculate the width of each subinterval: \( \Delta x = \frac{3-1}{10} = 0.2 \).
• Determine the midpoint for each subinterval, given by \( x_i = 1 + 0.1 + i \cdot 0.2 \) for \( i = 0, 1, \ldots, 9 \).
• Evaluate the function \( e^{-2x} \) at each midpoint and sum up the results: \( M_{10} = 0.2 \sum_{i=0}^{9} e^{-2(1.2 + 0.2i)} \).

This computation gives us an approximate value for the integral. Compared to the exact value, this method provided a close approximation with a very small error of 0.00007.

The Trapezoidal Rule is another numerical integration method that approximates the area under a curve by summing up trapezoids rather than rectangles. It can be more accurate than the midpoint rule because it incorporates the slope of the function in each subinterval by averaging the function values at the subinterval endpoints.

For the integral \( \int_{1}^{3} e^{-2x} \, dx \) using the trapezoidal rule:

• The interval is divided into 10 subintervals with \( \Delta x = 0.2 \).
• The formula is \( T_{10} = \frac{\Delta x}{2} \left[ f(x_0) + 2\sum_{i=1}^{9} f(x_i) + f(x_{10}) \right] \).
• Plug in the endpoint values of each interval to compute: \( T_{10} \approx 0.0528 \).

This method achieved an approximate result very close to the exact value. The error is slightly larger than the midpoint approximation error but remains minor, at 0.00017.

Simpson's Rule leverages quadratic polynomials to approximate the integral, offering higher accuracy than the previous methods when dealing with smooth, continuous functions. It combines the midpoint and trapezoidal methods and adjusts for the curvature of the function with a weighted average.

For our integral \( \int_{1}^{3} e^{-2x} \, dx \), Simpson’s Rule gives astonishing precision:

• Divide the interval into 20 subintervals, yielding \( \Delta x = 0.1 \).
• Utilize the formula: \( S_{20} = \frac{\Delta x}{3} \left[ f(x_0) + 4 \sum_{i=1,3,\ldots,19} f(x_i) + 2\sum_{i=2,4,\ldots,18} f(x_i) + f(x_{20}) \right] \).
• Calculate the sum using values at odd and even index points to achieve \( S_{20} \approx 0.0527 \).

Simpson's Rule managed to closely match the exact integral value with a minimal error of 0.00007, highlighting its efficiency for this type of function.