Shark Displacement Calculation Using Trapezoidal Rule

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In this article, we will delve into the fascinating world of shark displacement speed and explore how to calculate it using the trapezoidal rule. Sharks, as apex predators of the ocean, exhibit remarkable swimming abilities, and understanding their speed is crucial for marine biologists and researchers. The provided function, v(t) = 36 - t², gives the displacement speed of a shark in kilometers per hour, where t represents time. Our goal is to estimate the shark's displacement over a specific time interval using the trapezoidal rule, a numerical integration technique. We will partition the interval [0, 6] into subintervals [0, 1], [1, 3], and [3, 6] and apply the trapezoidal rule to each subinterval. This method allows us to approximate the definite integral of the velocity function, which represents the total displacement of the shark during the given time period. The trapezoidal rule is particularly useful when an analytical solution to the integral is difficult or impossible to find, making it a valuable tool in various scientific and engineering applications. By the end of this article, you will have a comprehensive understanding of how to apply the trapezoidal rule to calculate displacement speed and appreciate its significance in real-world scenarios.

Understanding the Problem

The problem at hand involves calculating the displacement of a shark given its velocity function, v(t) = 36 - t², over the time interval [0, 6]. The velocity function describes how the shark's speed changes with time, and displacement refers to the shark's overall change in position during this period. To solve this, we will employ the trapezoidal rule, a numerical method used to approximate the definite integral of a function. In simpler terms, the definite integral of the velocity function over a time interval gives us the displacement. Since we are provided with the velocity function v(t) = 36 - t², we could potentially find the exact displacement by analytically integrating this function from 0 to 6. However, the problem specifically instructs us to use the trapezoidal rule with a given partition, which provides an opportunity to understand and apply this important numerical technique.

The trapezoidal rule works by dividing the area under the curve of the function into trapezoids and summing their areas to approximate the total area under the curve. This method is particularly useful when dealing with functions that are difficult to integrate analytically or when only discrete data points are available. The partition provided, [0, 1] ∪ [1, 3] ∪ [3, 6], divides the time interval [0, 6] into three subintervals. We will apply the trapezoidal rule to each subinterval separately and then sum the results to obtain the total approximate displacement. This step-by-step approach allows for a more accurate approximation compared to using a single trapezoid over the entire interval. Furthermore, understanding how to apply the trapezoidal rule in this context provides a valuable skill applicable to various problems in physics, engineering, and other fields where numerical integration is necessary.

Trapezoidal Rule Explained

The trapezoidal rule is a numerical integration technique used to approximate the definite integral of a function. It works by dividing the area under the curve of the function into trapezoids and summing their areas. This method is particularly useful when finding the exact integral is difficult or impossible, or when we only have discrete data points. To understand the trapezoidal rule, let's first consider the basic idea behind numerical integration. The definite integral of a function f(x) from a to b represents the area under the curve of f(x) between the vertical lines x = a and x = b. The trapezoidal rule approximates this area by dividing the interval [a, b] into n subintervals and constructing trapezoids within each subinterval.

The formula for the trapezoidal rule is derived from the formula for the area of a trapezoid, which is given by (1/2) * height * (base1 + base2). In the context of numerical integration, the height of the trapezoid corresponds to the width of the subinterval, and the bases correspond to the function values at the endpoints of the subinterval. The trapezoidal rule formula for approximating the definite integral ∫[a,b] f(x) dx is given by: (Δx/2) * [f(x₀) + 2f(x₁) + 2f(x₂) + ... + 2f(xₙ₋₁) + f(xₙ)], where Δx is the width of each subinterval, and x₀, x₁, ..., xₙ are the endpoints of the subintervals. The term Δx is calculated as (b - a) / n, where n is the number of subintervals. This formula essentially calculates the weighted sum of the function values at the endpoints of the subintervals, with the interior points being weighted twice as much as the endpoints. The factor of 1/2 in the formula arises from the fact that each trapezoid covers half the width of the subinterval. The trapezoidal rule is an important tool in various fields, including engineering, physics, and computer science, where it is used to approximate integrals that cannot be easily solved analytically. It provides a balance between accuracy and computational complexity, making it a versatile method for numerical integration.

Applying the Trapezoidal Rule to the Problem

To apply the trapezoidal rule to the given problem, we need to calculate the displacement of the shark using the velocity function v(t) = 36 - t² over the interval [0, 6] with the partition [0, 1] ∪ [1, 3] ∪ [3, 6]. This means we will apply the trapezoidal rule to each subinterval separately and then sum the results to obtain the total approximate displacement. First, let's consider the subinterval [0, 1]. The width of this subinterval is Δt₁ = 1 - 0 = 1. We need to evaluate the function v(t) at the endpoints of the subinterval, which are t = 0 and t = 1. So, v(0) = 36 - 0² = 36 and v(1) = 36 - 1² = 35. Applying the trapezoidal rule formula for this subinterval, we get: Area₁ ≈ (Δt₁ / 2) * [v(0) + v(1)] = (1 / 2) * [36 + 35] = 0.5 * 71 = 35.5. This represents the approximate displacement of the shark during the time interval [0, 1].

Next, we consider the subinterval [1, 3]. The width of this subinterval is Δt₂ = 3 - 1 = 2. We evaluate the function v(t) at the endpoints t = 1 and t = 3. We already know v(1) = 35, and v(3) = 36 - 3² = 36 - 9 = 27. Applying the trapezoidal rule formula for this subinterval, we get: Area₂ ≈ (Δt₂ / 2) * [v(1) + v(3)] = (2 / 2) * [35 + 27] = 1 * 62 = 62. This represents the approximate displacement of the shark during the time interval [1, 3]. Finally, we consider the subinterval [3, 6]. The width of this subinterval is Δt₃ = 6 - 3 = 3. We evaluate the function v(t) at the endpoints t = 3 and t = 6. We already know v(3) = 27, and v(6) = 36 - 6² = 36 - 36 = 0. Applying the trapezoidal rule formula for this subinterval, we get: Area₃ ≈ (Δt₃ / 2) * [v(3) + v(6)] = (3 / 2) * [27 + 0] = 1.5 * 27 = 40.5. This represents the approximate displacement of the shark during the time interval [3, 6]. To find the total approximate displacement over the entire interval [0, 6], we sum the areas calculated for each subinterval: Total Displacement ≈ Area₁ + Area₂ + Area₃ = 35.5 + 62 + 40.5 = 138. Therefore, the approximate displacement of the shark over the time interval [0, 6] is 138 kilometers.

Step-by-Step Calculation

To provide a clear and concise understanding of the calculation process, let's break down the step-by-step application of the trapezoidal rule to the given problem. This will help to illustrate the methodology and ensure accuracy in the calculations. The problem requires us to estimate the displacement of a shark using the velocity function v(t) = 36 - t² over the time interval [0, 6] with the partition [0, 1] ∪ [1, 3] ∪ [3, 6]. This means we need to apply the trapezoidal rule to each subinterval separately and then sum the results. Here's a detailed breakdown of each step:

  1. Identify the Subintervals: The given partition divides the interval [0, 6] into three subintervals: [0, 1], [1, 3], and [3, 6]. We will apply the trapezoidal rule to each of these subintervals.
  2. Calculate the Width of Each Subinterval: The width of a subinterval, denoted as Δt, is the difference between the endpoints of the subinterval. For the first subinterval [0, 1], Δt₁ = 1 - 0 = 1. For the second subinterval [1, 3], Δt₂ = 3 - 1 = 2. For the third subinterval [3, 6], Δt₃ = 6 - 3 = 3.
  3. Evaluate the Function at the Endpoints of Each Subinterval: We need to evaluate the velocity function v(t) = 36 - t² at the endpoints of each subinterval.
    • For the subinterval [0, 1]: v(0) = 36 - 0² = 36 and v(1) = 36 - 1² = 35.
    • For the subinterval [1, 3]: v(1) = 35 (already calculated) and v(3) = 36 - 3² = 36 - 9 = 27.
    • For the subinterval [3, 6]: v(3) = 27 (already calculated) and v(6) = 36 - 6² = 36 - 36 = 0.
  4. Apply the Trapezoidal Rule Formula to Each Subinterval: The trapezoidal rule formula for a subinterval [a, b] is given by: Area ≈ (Δt / 2) * [v(a) + v(b)].
    • For the subinterval [0, 1]: Area₁ ≈ (1 / 2) * [36 + 35] = 0.5 * 71 = 35.5.
    • For the subinterval [1, 3]: Area₂ ≈ (2 / 2) * [35 + 27] = 1 * 62 = 62.
    • For the subinterval [3, 6]: Area₃ ≈ (3 / 2) * [27 + 0] = 1.5 * 27 = 40.5.
  5. Sum the Areas to Find the Total Approximate Displacement: The total approximate displacement is the sum of the areas calculated for each subinterval: Total Displacement ≈ Area₁ + Area₂ + Area₃ = 35.5 + 62 + 40.5 = 138. Therefore, the approximate displacement of the shark over the time interval [0, 6] is 138 kilometers.

Conclusion

In conclusion, we have successfully applied the trapezoidal rule to estimate the displacement of a shark using the velocity function v(t) = 36 - t² over the time interval [0, 6] with the partition [0, 1] ∪ [1, 3] ∪ [3, 6]. By dividing the interval into subintervals and approximating the area under the curve using trapezoids, we were able to estimate the shark's displacement with reasonable accuracy. The step-by-step calculation demonstrated the practical application of the trapezoidal rule, highlighting its utility in scenarios where analytical integration is complex or impossible. The trapezoidal rule provides a valuable tool for approximating definite integrals and is widely used in various fields, including physics, engineering, and computer science. This method allows us to make estimations and gain insights into real-world problems, such as the movement of marine animals, even when precise analytical solutions are not readily available. The final result of 138 kilometers represents the approximate total displacement of the shark during the given time period.

The exercise of calculating shark displacement speed using the trapezoidal rule not only reinforces the understanding of numerical integration techniques but also underscores the importance of mathematical modeling in understanding natural phenomena. The velocity function v(t) = 36 - t² provides a simplified representation of a shark's movement, and the trapezoidal rule allows us to estimate its overall displacement based on this model. While the trapezoidal rule provides an approximation, its accuracy can be improved by increasing the number of subintervals, thereby reducing the width of each trapezoid and better fitting the curve of the function. In real-world applications, more complex models and numerical methods may be employed to achieve higher precision, but the fundamental principles remain the same. This exploration into shark displacement speed serves as an excellent example of how mathematical tools can be used to analyze and interpret the world around us, fostering a deeper appreciation for the interplay between mathematics and natural sciences. The ability to apply numerical methods like the trapezoidal rule is a valuable skill for students and professionals alike, enabling them to tackle a wide range of problems in diverse fields.