Consider the given function.
f(x) = e<sup>x</sup> - 6
If f(x) = e<sup>x</sup> - 6, 0 ? x ? 2, find the Riemann sum with n = 4 correct to six decimal places, taking the sample points to be midpoints.
Step 1
We must calculate $M_4 = \sum_{i=1}^{4} f(\bar{x_i}) \Delta x = [f(\bar{x_1}) + f(\bar{x_2}) + f(\bar{x_3}) + f(\bar{x_4})] \Delta x$, where $\bar{x_1}$, $\bar{x_2}$, $\bar{x_3}$, and $\bar{x_4}$ represent the midpoints of four equal sub-intervals of [0, 2].
Since we wish to estimate the area over the interval [0, 2] using 4 rectangles of equal widths, then each rectangle will have width $\Delta x = $