Problem 9 - Applied Calculus

Question

Use the following table to estimate $$\int_{3}^{4} W(t) d t .$$ What are \(n\) and \(\Delta t ?\) $$\begin{array}{c|c|c|c|c|c|c} \hline t & 3.0 & 3.2 & 3.4 & 3.6 & 3.8 & 4.0 \\\\\hline W(t) & 25 & 23 & 20 & 15 & 9 & 2 \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

\(n = 5\), \(\Delta t = 0.2\), and the integral is approximately 16.1.

Step 1: Understanding the Problem

We need to estimate the integral \(\int_{3}^{4} W(t) \, dt\) using the provided table of values for \(W(t)\). This involves approximating the area under the curve \(W(t)\) from \(t = 3\) to \(t = 4\).

Step 2: Identify the Method

To estimate the integral, we can use numerical integration methods like the Trapezoidal Rule, which involves summing up areas of trapezoids under the curve. The choice of method may depend on the regularity and spacing of the data points.

Step 3: Find \(n\) and \(\Delta t\)

From the table, there are 6 data points, corresponding to \(t = 3.0, 3.2, ..., 4.0\). Therefore, \(n\) is the number of subdivisions, which is 5 (since \(n\) is the number of intervals), and \(\Delta t = 3.2 - 3.0 = 0.2\), the uniform spacing between the data points.

Step 4: Applying the Trapezoidal Rule

The Trapezoidal Rule formula is given by: \[\int_{a}^{b} W(t) \, dt \approx \frac{\Delta t}{2}\left( W(t_0) + 2 \sum_{i=1}^{n-1} W(t_i) + W(t_n) \right)\] In our case, \(t_0 = 3\), \(t_1 = 3.2\), ..., \(t_5 = 4\); thus, the estimation is: \[\frac{0.2}{2}(25 + 2(23 + 20 + 15 + 9) + 2) = 0.1(25 + 134 + 2) = 16.1\]

Step 5: Calculating Sum Inside Trapezoidal Rule

Calculate the sum: \(W(t_1) + W(t_2) + W(t_3) + W(t_4) = 23 + 20 + 15 + 9 = 67\). Thus, \[25 + 134 + 2 = 161\].

Step 6: Final Calculations

Using the final value from the sum, Trapezoidal estimation: \[0.1 \times 161 = 16.1\].

Key Concepts

Trapezoidal Rule

When we talk about numerical integration, one popular method is the Trapezoidal Rule. This rule helps us estimate the area under a curve by approximating it with trapezoids. The idea is to divide the area into small, manageable sections and calculate the area of each. Then, we sum up each of these areas.
Here's how it works:
- Each pair of consecutive points on a curve is connected with a straight line, forming a trapezoid.
- The height of each trapezoid is the difference between two consecutive points, often called \( \Delta t \).
- The area of each trapezoid is calculated and then added together to obtain the total area under the curve.
The formula for the Trapezoidal Rule is: \[ \int_{a}^{b} f(x) \, dx \approx \frac{\Delta t}{2}\left( f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n) \right) \] This allows us to get a more precise approximation of the integral by increasing the number of subdivisions (\( n \)). Remember, the more trapezoids we use, the more accurate the approximation becomes.

Approximate Integrals

In calculus, integrals are often expressed with exact values. However, there are situations where finding an exact integral is difficult or the data is only available at discrete points. This is where approximate integrals come into play.
Approximate integrals compute the area under a curve by breaking it down into simpler segments, like straight lines, rather than attempting to calculate the precise curve. This approach is particularly helpful when dealing with:
- Irregular data points,
- Complex functions, or
- Real-world data that do not adhere to neat, analytical forms
One should be aware that such approximations provide only an estimate of the actual area. The precision of these integrals depends heavily on the number of data points and methods used, such as the Trapezoidal Rule or Simpson's Rule. It's crucial to understand that while these are powerful tools, they won't always provide the exact answer but a very useful approximation.

Data Tables in Calculus

Data tables are invaluable in calculus for performing numerical approximations, especially for functions whose exact expressions are unknown. By using known values of a function at certain points, these tables allow us to estimate integrals over an interval:
A typical data table provides:
- The values of the independent variable such as \( t \),
- The corresponding values of the dependent function \( W(t) \) at those points.
These values form the basis for numerical strategies like the Trapezoidal Rule. The table allows you to determine the number of intervals \( n \) as well as the spacing \( \Delta t \) between the data points, which are crucial for calculating the area under the curve using numerical methods.
In summary, data tables consolidate complex real-world data into manageable chunks, making the problem of integration far more approachable. This makes them a preferred tool in numerous applications, from physics to economics, where data isn't continuous or straightforward from the start.