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The Midpoint Rule is a numerical method for approximating definite integrals. It works by dividing the area under a curve into a series of rectangles, whose height is determined by the function value at the midpoint of each subinterval. This method is particularly useful when you want an estimate using equally spaced intervals.

For our example, we divided the interval from 0 to 1 into 50 subintervals, each with a width of \(h = 0.02\). The midpoints are at \(x_i = 0.01, 0.03, 0.05, \ldots, 0.99\). By calculating the function value \(e^{-x_i}\) at each midpoint and summing these values multiplied by the subinterval width \(h\), we get the Midpoint Rule approximation. This produces a result close to the actual integral, with our example yielding approximately 0.632275. This approach gives a good estimate, especially when more subintervals are used.

The Trapezoid Rule is another method for approximating definite integrals, which employs trapezoids instead of rectangles. This technique assumes the function is linear between consecutive points and calculates the integral as a sum of areas of trapezoids formed under the curve.

In our exercise, we also used 50 subintervals. The function is evaluated at the endpoints, forming points \(x_0 = 0\) and \(x_{50} = 1\), and the approximate area under the curve is calculated using the formula:

• \(T_{50} = \frac{0.02}{2} \left[e^0 + 2\sum_{i=1}^{49} e^{-x_i} + e^{-1}\right]\)

The Trapezoid Rule thus provides an approximation of 0.632085 for the integral. This method can often be more accurate than the Midpoint Rule, especially with a finer subinterval split.

A definite integral computes the accumulation of values of a function over a specified interval. Essentially, it finds the net area under a curve, which can represent various physical quantities like distance, area, or volume in real-life scenarios.

In our problem, we find the definite integral of \(e^{-x}\) from 0 to 1. The exact solution can be found by determining the antiderivative, which for \(e^{-x}\) is \(-e^{-x}\). Evaluating this from 1 to 0 gives us the exact integral value, \(1 - e^{-1} \approx 0.632120\). Understanding definite integrals is central to many applications in mathematics and science, providing exact measurements where necessary.

Relative error helps gauge the accuracy of an approximation by comparing it to the exact value. It gives an idea of how close the numerical approximation is to the true value of the integral.

Calculating the relative error involves finding the difference between the approximate value and the exact integral, divided by the exact value:

• Relative Error for Midpoint Rule: \(\frac{|M_{50} - \int_{0}^{1} e^{-x} dx|}{\int_{0}^{1} e^{-x} dx}\)
• Relative Error for Trapezoid Rule: \(\frac{|T_{50} - \int_{0}^{1} e^{-x} dx|}{\int_{0}^{1} e^{-x} dx}\)

For the Midpoint Rule, the relative error calculated is approximately 0.000245, and for the Trapezoid Rule, it is approximately 0.000056. These numbers indicate the Trapezoid Rule approximation is slightly closer to the true value, showing how relative error can guide improvements in numerical methods.