Problem 6 - Applied Calculus

Question

Use the following table to estimate $$\int_{10}^{26} f(x) d x$$. $$\begin{array}{c|c|c|c|c|c} \hline x & 10 & 14 & 18 & 22 & 26 \\\\\hline f(x) & 100 & 88 & 72 & 50 & 28 \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

The estimated integral is 1104.

Step 1: Understand the Trapezoidal Rule

The trapezoidal rule is a numerical method used to estimate the value of a definite integral. It approximates the area under a curve by dividing it into trapezoids rather than rectangles. The formula for the trapezoidal rule is: \( \int_{a}^{b} f(x) \, dx \approx \frac{h}{2} \left[y_0 + 2(y_1 + y_2 + ... + y_{n-1}) + y_n\right] \), where \( h \) is the width of each sub-interval.

Step 2: Calculate Sub-interval Width

First, determine the width \( h \) of each sub-interval. In our case, since the points are equally spaced, \( h \) is the difference between consecutive \( x \)-values, which is \( h = 14 - 10 = 4 \).

Step 3: Extract Values for Calculation

Let \( y_0 = f(10) = 100 \), \( y_1 = f(14) = 88 \), \( y_2 = f(18) = 72 \), \( y_3 = f(22) = 50 \), and \( y_4 = f(26) = 28 \). These are the \( y \)-values corresponding to each \( x \)-value from the table.

Step 4: Apply Trapezoidal Rule

Substitute the values into the trapezoidal rule formula: \[ \int_{10}^{26} f(x) \, dx \approx \frac{4}{2} \left[100 + 2(88 + 72 + 50) + 28 \right] \].

Step 5: Simplify the Expression

Carry out the arithmetic calculations within the brackets: \( 100 + 2(88 + 72 + 50) + 28 = 100 + 424 + 28 = 552 \).

Step 6: Compute Final Estimate

Calculate the final estimate: \[ \frac{4}{2} \times 552 = 2 \times 552 = 1104 \]. This is the estimated value of the integral.

Key Concepts

Numerical Integration

Numerical integration is a crucial technique in mathematics used to approximate the definite integral of a function. Instead of finding an exact solution, which may be complex or even impossible for certain functions, numerical methods provide an estimated solution. There are various ways to perform numerical integration; some popular methods include:

• Trapezoidal Rule
• Simpson's Rule
• Midpoint Rule

Each method involves different levels of approximation and applies better to some functions than others. The basic idea is to replace the area under a curve with a shape that is easier to compute, like a trapezoid or a rectangle.
Regardless of the shape chosen, numerical integration simplifies the process of finding the area under the curve, especially when working with irregular functions or data points without an analytical expression.
Using numerical integration techniques, especially the Trapezoidal Rule, is particularly helpful in computing integrals from data in tables, where the function is not explicitly given.

Definite Integral Estimation

Definite integrals represent the area under the curve of a function, from one point to another, on the x-axis. In many real-world problems, calculating these definite integrals directly can be challenging. This is where estimation methods come in handy.
The Trapezoidal Rule, one of the most common estimation methods, is used to approximate the value of a definite integral. It breaks the area into small trapezoidal slices and calculates the area of each.
To apply the Trapezoidal Rule, you need to know:

• The values of the function at specific points (these become the vertices of the trapezoids).
• The width of the intervals (spacing between these points).

In our exercise, the function values and intervals were neatly provided in a table, making it straightforward to apply the Trapezoidal Rule.
By summing up the areas of the trapezoids, the total area under the curve, which estimates the definite integral, is quickly obtained.

Applied Calculus

Applied Calculus involves using calculus concepts to solve practical problems in various fields such as physics, engineering, economics, and even biology. One of its key components is the concept of integration, particularly for solving real-life problems where analytic solutions are hard to find.
The estimation of definite integrals using numerical methods, such as the Trapezoidal Rule, is a classic example of applied calculus. These methods turn abstract mathematical concepts into concrete solutions for practical issues.
Applications of such integration could include:

• Computing the total distance traveled by an object given velocity data at discrete time intervals.
• Estimating the area under curves that model economic trends, without having an explicit function.
• Agricultural and biological applications where the measurement of growth or surface area over time is needed.

Using numerical methods brings the power of calculus to scenarios where direct solutions were previously inaccessible. This approach allows for approximations that are sufficiently accurate for practical needs.