8. Definite Integrals

Sigma notation Express the following sums using sigma notation. (Answers are not unique.)

(d) 1 + 1/2 + 1/3 + 1/4

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                         

Sigma notation Evaluate the following expressions.                                                                                                                                          

  (f)      3                                                                                                                                                                               

       ∑ (3j ― 4)                                                                                                                                                                          

      j =1                         

Displacement from a table of velocities The velocities (in mi/hr) of an automobile moving along a straight highway over a two-hour period are given in the following table.

(b) Find the midpoint Riemann sum approximation to the displacement on [0,2] with n = 2 and .n = 4 .

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

41-44. {Use of Tech} Nonuniform grids

Use the indicated methods to solve the following problems with nonuniform grids.

41. A curling iron is plugged into an outlet at time t = 0. Its temperature T in degrees Fahrenheit, assumed to be a continuous function that is strictly increasing and concave down on 0 ≤ t ≤ 120, is given at various times (in seconds) in the table.

a. Approximate (1/120)∫(0 to 120)T(t)dt in three ways using a left Riemann sum, using a right Riemann sum and using the Trapezoid Rule

Interpret the value of (1/120)∫(0 to 120)T(t)dt in the context of this problem.

42. Approximating integrals The function f is twice differentiable on (-∞, ∞). Values of f at various points on [0, 20] are given in the table.

a. Approximate ∫(0 to 120) f(x) dx in three way using a left Riemann sum, a right Riemann sum and the Trapezoid Rule

A piece of wood paneling must be cut in the shape shown in the figure.

The coordinates of several points on its curved surface are also shown (with units of inches).

a. Estimate the surface area of the paneling using the Trapezoid Rule.

45–48. {Use of Tech} Trapezoid Rule and Simpson’s Rule Consider the following integrals and the given values of n.

45. ∫(0 to 1) e^(2x) dx; n = 25

c. Compute the absolute errors in the Trapezoid Rule and Simpson’s Rule with 2n subintervals.