Ch. 8 - Techniques of Integration

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 8 - Techniques of Integration

Problema 8.7.23

Capitolo 8, Problema 8.7.23

Volume of water in a swimming pool
A rectangular swimming pool is 30 ft wide and 50 ft long. The accompanying table shows the depth h(x) of the water at 5-ft intervals from one end of the pool to the other. Estimate the volume of water in the pool using the Trapezoidal Rule with n = 10 applied to the integral
V = ∫ from 0 to 50 of 30 · h(x) dx.

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Identify the integral to estimate the volume of water in the pool: \(V = \int_0^{50} 30 \cdot h(x) \, dx\), where 30 ft is the width of the pool and \(h(x)\) is the depth at position \(x\).

Note that the depth values \(h(x)\) are given at 5-ft intervals from \(x=0\) to \(x=50\), so the interval width is \(\Delta x = 5\) ft, and the number of subintervals is \(n=10\).

Apply the Trapezoidal Rule formula for \(n=10\) subintervals: \(\int_a^b f(x) \, dx \approx \frac{\Delta x}{2} \left[f(x_0) + 2f(x_1) + 2f(x_2) + \cdots + 2f(x_{n-1}) + f(x_n)\right]\), where \(f(x) = 30 \cdot h(x)\) in this problem.

Calculate \(f(x_i) = 30 \times h(x_i)\) for each depth value \(h(x_i)\) from the table, then substitute these values into the trapezoidal sum expression.

Finally, multiply the sum by \(\frac{\Delta x}{2} = \frac{5}{2}\) to estimate the volume \(V\). This will give the approximate volume of water in the pool.

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Definite Integral as Volume Calculation

The definite integral ∫ from a to b of a function represents the accumulation of quantities, such as area or volume. In this problem, the integral V = ∫ from 0 to 50 of 30 · h(x) dx calculates the volume of water by integrating the cross-sectional area (width times depth) along the pool's length.

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Percorso guidato

05:43

Definition of the Definite Integral

Trapezoidal Rule for Numerical Integration

The Trapezoidal Rule approximates the value of a definite integral by dividing the interval into subintervals and approximating the area under the curve as trapezoids. It is especially useful when the function values are known at discrete points, as in the given depth table.

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Percorso guidato

05:56

Additional Rules for Indefinite Integrals

Using Discrete Data Points for Approximation

When a function is given only at specific points, numerical methods like the Trapezoidal Rule use these discrete values to estimate integrals. Here, the depth h(x) is provided at 5-ft intervals, allowing the approximation of the integral and thus the volume of water.

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07:17

Linearization

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