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The Midpoint Rule is a numerical integration method used to estimate the value of a definite integral. It works by dividing the interval into equal segments and then calculating the area of rectangles using the function's value at the midpoint of each segment. This is useful when finding the area under a curvy graph.To approximate the integral \( \int_{1}^{3} e^{-2x} \, dx \) using the midpoint rule with 10 subintervals, we start by calculating the width of each interval, \( \Delta x = \frac{3-1}{10} = 0.2 \). We then find the midpoints \( x_i^* = 1 + (i-0.5)\Delta x \) for \( i = 1, 2, \ldots, 10 \).For each midpoint, compute the function value \( f(x_i^*) = e^{-2x_i^*} \), sum these values, and multiply by \( \Delta x \). This gives an approximation \( M_{10} \) of 0.0685. Although this may not be the exact value, it provides a good estimate. The absolute error here is given as \(|0.0676676 - 0.0685| \approx 0.0008324\). This error measure tells us how close our approximation is to the real value. The Midpoint Rule is often preferred for its accuracy with fewer subintervals when compared to some other basic numerical methods.

The Trapezoidal Rule is another classical method for numerical integration, which calculates the area under a curve by approximating it with a series of trapezoids. You essentially join consecutive points on the curve with straight line segments, forming trapezoid shapes.In this method, to approximate the integral \( \int_{1}^{3} e^{-2x} \, dx \), again, divide the interval from 1 to 3 into 10 equal parts, each with width \( \Delta x = 0.2 \). The formula used is:\[ T_n = \frac{\Delta x}{2} \left(f(a) + 2\sum_{i=1}^{n-1} f(x_i) + f(b) \right) \]Here, \( f(a) \) and \( f(b) \) are the function values at the endpoints of the integral, and the summation covers all the internal points. This results in a trapezoidal approximation \( T_{10} \approx 0.0694 \).The absolute error for this method is computed by the difference between the exact integral value and the trapezoidal approximation, resulting in \(|0.0676676 - 0.0694| \approx 0.0017324\). The Trapezoidal Rule provides a reasonably good estimate for integrals, though it tends to be less accurate than some other methods like Simpson's Rule, especially when the function curve is non-linear.

Simpson's Rule is a powerful numerical integration technique that provides accurate results by using parabolic arcs instead of straight lines to approximate the area under curves. This method is generally more precise than both the Midpoint and Trapezoidal rules when applied to the same number of subintervals.Simpson's Rule takes into account both the endpoints and midpoints over an even number of intervals. For the integral \( \int_{1}^{3} e^{-2x} \, dx \), with 20 subintervals (since Simpson's Rule requires an even number), each interval has width \( \Delta x = 0.1 \).The formula used is:\[ S_{2n} = \frac{\Delta x}{3} \left(f(a) + 4\sum_{i=1}^{n} f(x_{2i-1}) + 2\sum_{i=1}^{n-1} f(x_{2i}) + f(b) \right) \]Applying Simpson's Rule, we find \( S_{20} \approx 0.0676684 \). The absolute error is extremely small, calculated as \(|0.0676676 - 0.0676684| \approx 0.0000008\). This small error signifies the high accuracy of Simpson's Rule, making it a preferred choice for definite integrals where high precision is needed. Simpson's Rule often provides the best approximation in scenarios involving smooth and well-behaved functions like exponentials.