Question

River flow ratesThe 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

Accepted Answer

Step 1: Understand the problem. We are tasked with estimating the integral ∫(0 to 24) r(t) dt, which represents the total amount of water that flows by the town over a 24-hour period. We will use the Trapezoidal Rule and Simpson's Rule with n = 6 subdivisions. Step 2: Divide the interval [0, 24] into n = 6 equal subintervals. The width of each subinterval, denoted as h, is calculated as h = (24 - 0) / 6 = 4 hours. Step 3: Identify the values of r(t) at the endpoints and intermediate points of the subintervals. From the graph, approximate r(t) at t = 0, 4, 8, 12, 16, 20, and 24. These values will be used in both the Trapezoidal Rule and Simpson's Rule. Step 4: Apply the Trapezoidal Rule formula: ∫(a to b) f(x) dx ≈ (h/2) * [f(x₀) + 2f(x₁) + 2f(x₂) + ... + 2f(xₙ₋₁) + f(xₙ)], where x₀, x₁, ..., xₙ are the points in the interval. Substitute the values of r(t) and h into this formula. Step 5: Apply Simpson's Rule formula: ∫(a to b) f(x) dx ≈ (h/3) * [f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + ... + 4f(xₙ₋₁) + f(xₙ)], where x₀, x₁, ..., xₙ are the points in the interval. Substitute the values of r(t) and h into this formula.

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Concetti chiave

Definite Integral

Trapezoidal Rule

Simpson's Rule