Sigma notation Evaluate the following expressions.                                                                                                                                          
  (f)      3                                                                                                                                                                               
       ∑ (3j ― 4)                                                                                                                                                                          
      j =1                         

Sigma notation Evaluate the following expressions.

(a)    10                                                                                                                                                                               

       ∑ κ                                                                                                                                                                          

       κ=1                         

Sigma notation Evaluate the following expressions.                                                                                                                                          

(b)    10                                                                                                                                                                               

       ∑  (2κ + 1)                                                                                                                                                                          

       κ=1                         

Sigma notation Evaluate the following expressions.

(d)     5                                                                                                                                                                              

       ∑ (1 + n²)                                                                                                                                                                          

       n=1                         

Sigma notation Evaluate the following expressions.                                                                                                                                          

(e)     3                                                                                                                                                                               

       ∑  (2m + 2) / 3                                                                                                                                                                          

      m =1                         

27. {Use of Tech} Midpoint Rule, Trapezoid Rule, and relative error

Find the Midpoint and Trapezoid Rule approximations to ∫(0 to 1) sin(πx) dx using n = 25 subintervals. Compute the relative error of each approximation.

29-34. {Use of Tech} Comparing the Midpoint and Trapezoid Rules

Apply the Midpoint and Trapezoid Rules to the following integrals. Make a table similar to Table 8.5 showing the approximations and errors for n = 4, 8, 16, and 32. The exact values of the integrals are given for computing the error.

30. ∫(0 to 6) (x³/16 - x) dx = 4

29-34. {Use of Tech} Comparing the Midpoint and Trapezoid Rules

Apply the Midpoint and Trapezoid Rules to the following integrals. Make a table similar to Table 8.5 showing the approximations and errors for n = 4, 8, 16, and 32. The exact values of the integrals are given for computing the error.

33. ∫(0 to π) sin x cos(3x) dx = 0

River flow rates

The following figure shows the discharge rate r(t) of the Snoqualmie River near Carnation, Washington, starting on a February day when the air temperature was rising. The variable t is the number of hours after midnight, r(t) is given in millions of cubic feet per hour, and ∫(0 to 24) r(t) dt equals the total amount of water that flows by the town of Carnation over a 24-hour period. Estimate ∫(0 to 24) r(t) dt using the Trapezoidal Rule and Simpson's Rule with the following values of n.

n = 6