This is my sixth question

6. (Optional Problem - 5 pts extra credit on this assignment - A shortcut to Simpson's Approximation.) In this problem you'll find a simple formula for finding the Simpson's Rule approximation. For people who frequently need to find approximate integrals based on data, this can be a very useful shortcut. Partition [a,b] into n equal pieces of length Ar = 6-9 the = atk(- ) for k = 0,1, ..., 2n, where Ar = b - a n Let yx = f(tx) for k = 0, ..., 2n. Ar Ar Ar AT . . . In to . . . 127 -4 In this exercise, use the labeling system described above. (a) Show that using this labeling convention that the trapezoid approximation with n slices is: In = 7 30 + 2y2 + 2ys + 2y6 + ...+ 2yan-2 + yan Ar. (b) Show that using this labeling convention that the midpoint approximation with n slices is: Mn = (31 + 93 + ys + yr + ... + yzn-1) Ar. (c) Remember that Simpson's Rule is a certain weighted average of the trapezoid and midpoint rules. Show that using the labeling convention above Simpson's Rule for n slices (that is, roughly 2n data points!) is given by: San = = (30 + 4y1 + 242 + 4y3 + 2ya + ... + 2yan-2 + 4y20-1 + 12n ) Ax ( 2 ) Notice that because Simpson's Rule requires both the midpoint and trapezoidal rules and the relevant data points for each of these two rules are different, Simpson's Rule requires approximately double the number of data points necessary for computing either M, or In individually, but it's surprisingly accurate. (d) A surveyor is measuring the cross-sectional area of a 30-foot-wide river beneath a bridge. He measures the river's depth every 5 feet. The data are given below. 5 ft 5 ft 5 ft 5 ft 5 ft Depth (in feet) 0.5 1.5 (2) The answer to part (c) is given at the end of the section in 26.2. Use the shortcut formula above for Simpson's Rule to approximate the cross-sectional area of the river. [Pro Tip: Think carefully about your value of Ar.]