Problem 3 - Applied Calculus

Question

Use the following table to estimate \(\int_{0}^{25} f(x) d x\). $$ \begin{array}{c|c|c|c|c|c|c} \hline x & 0 & 5 & 10 & 15 & 20 & 25 \\ \hline f(x) & 100 & 82 & 69 & 60 & 53 & 49 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The estimated value of the integral is 1692.5.

Step 1: Understand the Trapezoidal Rule

The Trapezoidal Rule is a numerical method to approximate the definite integral of a function. It estimates the area under the curve as a series of trapezoids. If we have function values at evenly spaced intervals, the approximation is calculated using the formula: \( T = \frac{b-a}{n} \left[ \frac{f(x_0) + f(x_n)}{2} + f(x_1) + f(x_2) + ... + f(x_{n-1}) \right] \), where \( a \) and \( b \) are the integral limits, and \( n \) is the number of sub-intervals.

Step 2: Identify Interval Width and Segments

We are given the function values at intervals of 5: 0, 5, 10, 15, 20, and 25. Thus, the interval width \( h \) is 5 and the number of intervals \( n \) is 5. These intervals divide our domain from 0 to 25 into 5 segments.

Step 3: Apply the Trapezoidal Rule Formula

Using the Trapezoidal Rule formula: \[\int_{0}^{25} f(x) \, dx \approx \frac{5}{2} \times (f(0) + 2f(5) + 2f(10) + 2f(15) + 2f(20) + f(25))\]Substitute the given values of \( f(x) \) into the formula.

Step 4: Substitute and Calculate

Substitute the provided function values into the formula:\[= \frac{5}{2} \times (100 + 2 \times 82 + 2 \times 69 + 2 \times 60 + 2 \times 53 + 49)\]Calculate:\[= \frac{5}{2} \times (100 + 164 + 138 + 120 + 106 + 49)\]

Step 5: Compute Final Result

Evaluate the expression inside the parentheses:\[100 + 164 + 138 + 120 + 106 + 49 = 677\].Now multiply by \( \frac{5}{2} \):\[\frac{5}{2} \times 677 = 1692.5\]. Thus, the estimated integral is 1692.5.

Key Concepts

Numerical Integration

Numerical integration is a technique used to approximate the value of a definite integral. When dealing with complex functions for which finding an exact integral is challenging or impossible, numerical methods become essential.
These methods, like the Trapezoidal Rule and Simpson’s Rule, allow us to compute an approximate area under the curve described by the function.
The Trapezoidal Rule is particularly useful when you have discrete data points, as it estimates the area by connecting these points in straight lines, forming trapezoids.

• Utilizes known function values at distinct intervals.
• Ideal for functions with easily computable or tabulated points.

Essentially, numerical integration interprets the intricate details of a function’s graph as a series of simpler, linear segments to sum up the total area.

Definite Integral

A definite integral, denoted often as \( \int_{a}^{b} f(x) \, dx \), represents the net area between the function \( f(x) \) and the x-axis over the interval \([a, b]\).
This area can be positive or negative depending on whether the curve lies above or below the axis.
In many practical scenarios, such as modeling physical phenomena, definite integrals describe total quantities or changes over a specified range.

• Defined by specific upper and lower bounds: \( a \) and \( b \).
• Encapsulates the concept of accumulation, aggregation, or summation of infinitesimal data points.

When using numerical integration techniques like the Trapezoidal Rule, the task is to approximate this described area through sums of geometric shapes, making it manageable to calculate with available data.

Function Approximation

Function approximation, especially in the context of numerical integration, involves estimating a function’s behavior using simpler mathematical expressions or data interpolation.
In our exercise, the discrete points \( (x, f(x)) \) are provided, and our task is to approximate the integral despite not knowing the exact function form.
This approach is valuable in fields where data is collected in segments or where complex equations govern the system but are difficult to specify.

• Creates a continuous representation from discrete data points.
• Helpful in predictive modeling and analysis.

The trapezoidal rule is a straightforward illustration of function approximation, converting the undefined spaces between data points into definable geometric areas, ensuring a functional estimate that aligns closely with observed values.