Problem 20 - Applied Calculus

Question

Use the following table to estimate the area between \(f(x)\) and the \(x\) -axis on the interval \(0 \leq x \leq 20.\) $$\begin{array}{r|rrrrr}\hline x & 0 & 5 & 10 & 15 & 20 \\\\\hline f(x) & 15 & 18 & 20 & 16 & 12 \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

The estimated area is approximately 337.5 square units.

Step 1: Understand the Problem

We need to estimate the area under the curve given by the function values at discrete points from the provided table on the interval \(0 \leq x \leq 20\). The points given are \((0,15), (5,18), (10,20), (15,16), (20,12)\).

Step 2: Choose an Estimation Method

We will use the trapezoidal rule to estimate the area under the curve. This method is suitable for estimating the area using discrete points because it approximates the region under the graph as a series of trapezoids.

Step 3: Calculate the Widths of the Trapezoids

The width \(\Delta x\) of each sub-interval is uniform and equal to \(5\), since \((b-a)/n = (20-0)/4 = 5\).

Step 4: Apply the Trapezoidal Rule Formula

Using the trapezoidal rule formula for estimating area, which is: \[A \approx \frac{\Delta x}{2} [f(x_0) + 2f(x_1) + 2f(x_2) + 2f(x_3) + f(x_4)]\] Plug in the values from the table:- \(x_0 = 0, f(x_0) = 15\)- \(x_1 = 5, f(x_1) = 18\)- \(x_2 = 10, f(x_2) = 20\)- \(x_3 = 15, f(x_3) = 16\)- \(x_4 = 20, f(x_4) = 12\)

Step 5: Calculate the Estimated Area

Substitute the function values and step size into the trapezoidal rule formula:\[A \approx \frac{5}{2} \left(15 + 2(18) + 2(20) + 2(16) + 12\right)\]- \(15 + 36 + 40 + 32 + 12 = 135\)Therefore,\[A \approx \frac{5}{2} \times 135 = \frac{5 \times 135}{2} = 337.5\]

Step 6: Conclude with the Estimated Area

The estimated area between the function \(f(x)\) and the \(x\)-axis over the interval \(0 \leq x \leq 20\) is approximately 337.5 square units.

Key Concepts

Area Estimation

Area estimation is a crucial concept in mathematics when you want to find out how much space is covered between a curve and the x-axis over a given interval. For many real-world applications, like interpreting physical space or calculating workloads, knowing the exact area can be invaluable. However, when we can't easily calculate the actual area under a curve due to complexity or limited data points, we use estimation methods. In this case, you have discrete points from a table rather than a continuous function. These scenarios make exact area calculation tricky, so estimation becomes necessary, and it serves as a practical approximation of the true area. To simplify such calculations, depicted graphs are transformed into manageable geometric shapes like trapezoids, facilitating easier computation of areas.

Numerical Integration

Numerical integration is a mathematical technique used to approximate the integral of a function, which in simpler terms is finding the area under a curve. When functions are complex or given by discrete data points, numerical methods provide ways to estimate the integral without needing exact algebraic solutions. The trapezoidal rule is one popular method of numerical integration.It is especially helpful because:

• It handles functions that are not easily integrable with traditional methods.
• You only need function values at a few selected points rather than the entire curve.
• It is relatively simple and quick to apply, especially when data is scarce.

In application, numerical integration requires certain parameters: the interval you are evaluating (\[0 \leq x \leq 20\] in this case), and function values at specific points. By effectively summing up these small areas, numerical integration allows us to estimate the entire area accurately over the given interval.

Discrete Points

Working with discrete points is essential for situations where continuous data isn't available, like measurements taken over time or specific intervals. In the given exercise, we're provided with specific points on the interval between 0 and 20. These represent the function values at chosen x-values. Identifying these points is crucial as they form the basis for estimation in numerical integration methods such as the trapezoidal rule. Handling discrete data means:

• Recognizing that each point represents a snapshot of the function's behavior at a particular moment.
• Ensuring that the interval between each point is consistent, facilitating simpler calculations.
• Utilizing these data points effectively allows for meaningful approximation of the area under a curve without needing additional data.

Thus, discrete points serve as foundational markers, allowing the computation of approximate areas when exact values aren't available.