This question has several parts that must be completed sequentially. If you skip a part of the question, you will not receive any points for the skipped part, and you will not be able to come back to the skipped part. Given the integral below, do the following. 1 16 cos(x2) dx 0 Exercise (a) Find the approximations T4 and M4 for the given interval. Step 1 The Midpoint Rule says that b f(x) dx a ≈ Mn = Δx[f(x1) + f(x2) + ... + f(xn)] with Δx = b − a n . We need to estimate 1 16 cos(x2) dx 0 with n = 4 subintervals. For this, Δx = 1 − 0 4 4 = 1/4 1/4 . Step 2 Let x1 represents the midpoint of the first subinterval 0, 1 4 . The midpoint 0, 1 4 is x1 = 1 4 + 0 2 = 1/8 1/8 . Step 3 Therefore, M4 = 1 4 f 1 8 + f 3 8 + f 5 8 + f 7 8 = 14.542509 14.542509 (rounded to six decimal places). Step 4 The Trapezoidal Rule says that b f(x) dx a ≈ Tn = Δx 2 [f(x0) + 2f(x1) + ... + 2f(xn − 1) + f(xn)]. We again need to estimate 1 16 cos(x2) dx 0 with n = 4 subintervals. We have Δx 2 = 1/8 1/8 . Step 5 Therefore, T4 = 1 8 f(0) + 2f 1 4 + 2f 1 2 + 2f 3 4 + f(1) = 14.332142 14.332142 (rounded to six decimal places). Exercise (b)