Problem 15 - Basic Technical Mathematics with Calculus

Question

calculate the indicated areas. All data are accurate to at least two significant digits. Soundings taken across a river channel give the following depths with the corresponding distances from one shore. $$\begin{array}{l|l|r|r|r|r|r|r|r|r|r}\text {Distance (ft) } & 0 & 50 & 100 & 150 & 200 & 250 & 300 & 350 & 400 & 450 & 500 \\ \hline \text {Depth (ft) } & 5 & 12 & 17 & 21 & 22 & 25 & 26 & 16 & 10 & 8 & 0 \end{array}$$ Find the area of the cross section of the channel using Simpson's rule.

Step-by-Step Solution

Answer

The area of the channel cross-section is approximately 8216.67 square feet.

Step 1: Understand Simpson's Rule

Simpson's Rule is used to approximate the integral (area under the curve) when given data at discrete intervals. It is based on polynomial approximation of the function. The formula for Simpson's Rule is \( \frac{\Delta x}{3} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \ldots + 2f(x_{n-2}) + 4f(x_{n-1}) + f(x_n) \right] \), where \( \Delta x \) is the interval width.

Step 2: Identify Intervals and Interval Width

Examine the given distance data. Determine that the data are given at uniform intervals, with a distance of 50 feet between each pair of consecutive points. This makes \( \Delta x = 50 \) ft.

Step 3: Confirm Even Number of Intervals

Simpson's Rule requires an even number of intervals. Count the intervals by subtracting the number of points (11) by one, giving us 10 intervals. Since 10 is even, Simpson's Rule can be applied.

Step 4: Apply Simpson's Rule

Apply the formula, substituting the given depth values as \( f(x) \): \[ \frac{50}{3} \left[ 5 + 4(12) + 2(17) + 4(21) + 2(22) + 4(25) + 2(26) + 4(16) + 2(10) + 4(8) + 0 \right] \]

Step 5: Compute the Expression Inside Brackets

Calculate the expression inside the brackets: \( 5 + 48 + 34 + 84 + 44 + 100 + 52 + 64 + 20 + 32 + 0 = 493 \).

Step 6: Final Calculation with Simpson's Rule

Substitute back the calculated expression: \[ \frac{50}{3} \times 493 = 8216.67 \] square feet.

Key Concepts

Simpson's Rule

Simpson's Rule is a popular method for approximating definite integrals. It's especially useful when dealing with data that is collected at discrete intervals, like the depths of a river at various distances across a channel. Instead of calculating the exact area under a curve, which can be complex and time-consuming, Simpson's Rule provides a way to find an approximate area with a good degree of accuracy.

The rule works by fitting a series of parabolic curves between data points. These parabolas mimic the actual curve, providing a smooth approximation. The formula for Simpson's Rule is \( \frac{\Delta x}{3} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \ldots + 2f(x_{n-2}) + 4f(x_{n-1}) + f(x_n) \right] \). Here, \( \Delta x \) is the width of each interval. Alternating multipliers of 4 and 2 are applied to the values of the function at these points.

This means that every second interval is multiplied by 4, while others by 2, except the first and last points, which are multiplied by 1. The method requires an even number of intervals, which ensures the symmetry needed for parabolic approximation.

Numerical Integration

Numerical integration provides ways to calculate the area under curves when an analytical solution is difficult or impossible. In many real-world scenarios, especially in engineering and physics, you may encounter functions for which primitives are too complex or undefined. In such cases, numerical methods like Simpson's Rule become invaluable.

Numerical integration essentially quantifies the space beneath a curve by summing up small sections of the area. With Simpson's Rule, these sections are constructed as parabolic slices that give a closer estimation compared to linear approximation methods like the Trapezoidal Rule.

By using discrete data points as reference, numerical integration doesn't require the function's explicit equation, making it versatile for handling practical problems where the data originates from empirical or experimental methods.

Area Calculation

The area calculation in this exercise involves determining the cross-sectional area of a river channel. Given the depth readings at set distances across the channel, these measurements plot a series of points that can be connected to form a rough outline of the channel's profile.

To calculate the area, you approximate the shape of this outline using mathematical methods. Using Simpson's Rule, this involves summing the areas of parabolic sections, which emulate the natural curves of the channel's shape.

When applying the rule, you find the interval width, \( \Delta x \), from the distances provided and plug in the corresponding depth for each interval. The calculated width multiplies through the sum, as highlighted in the steps, to give a final area measurement in square feet.

Cross Section

A cross section is essentially a cut through a 3D object, exposing its internal configuration, or showing its outline at a point of interest. In this context, the cross section of a river channel provides a vertical slice of the river's profile from shore to shore.

Understanding a river's cross section is crucial for numerous applications, such as designing bridges, understanding flood risks, or studying sediment transport. It helps in visualizing the depth variations across the channel, and allows for calculations that model the river's flow and storage capacity.

Through measurements such as depths at given distances, you digitally reconstruct the cross-sectional area, facilitating analysis.

Mathematical Approximation

Mathematical approximation is essential when processes or shapes are too complex to derive exact solutions practically. In this exercise, the approximation comes into play through Simpson's Rule, used to estimate the area under a curve formed by uneven data points.

When using approximations, it's crucial to understand the trade-off between simplicity and accuracy. Techniques like Simpson's Rule enhance accuracy by incorporating the curve's natural undulations and providing a smoother estimate than linear methods.

These approximations enable the conversion of arbitrary experimental data into actionable insights, offering a bridge from raw numbers to clear, comprehensible analysis.