8. Definite Integrals

The length of one arch of the curve y = sin x is given by

L = ∫(from 0 to π) √(1 + cos²(x)) dx.

Estimate L by Simpson's Rule with n = 8.

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

{Use of Tech} Approximating net area The following functions are positive and negative on the given interval.

f(𝓍) = x³ on [-1,2]

(b) Approximate the net area bounded by the graph of f and the x-axis on the interval using a left, right, and midpoint Riemann sum with n = 4.

The instructions for the integrals in Exercises 1–10 have three parts, one for the Midpoint Rule, one for the Trapezoidal Rule, and one for Simpson’s Rule.

III. Using Simpson's Rule

a. Estimate the integral with n = 4 steps and find an upper bound for |ES|.

∫ from 0 to 2 of (t³ + t) dt

In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (b) Simpson’s Rule. (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)

∫ from -1 to 1 of (x² + 1) dx