Problem 48 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 43-48, use a graphing calculator to graph each integrand. Then use the Boundedness Property (Theorem \(C\) ) to find a lower bound and an upper bound for each definite integral. $$ \int_{0.2}^{0.4}\left(0.002+0.0001 \cos ^{2} x\right) d x $$

Step-by-Step Solution

Answer

The integral is bounded by 0.0004 and 0.00042.

Step 1: Graph the Integrand

Using a graphing calculator, graph the integrand function \( f(x) = 0.002 + 0.0001 \cos^2 x \) over the interval \([0.2, 0.4]\). This will help us visualize the behavior of the function in this domain.

Step 2: Identify Minimum and Maximum Values

From the graph, note the minimum and maximum values of the function \( f(x) \) on the interval \([0.2, 0.4]\). The function \( f(x) = 0.002 + 0.0001 \cos^2 x \) fluctuates slightly and typically remains very close to the value of 0.002 due to the small coefficient on the \( \cos^2 x \) term.

Step 3: Apply the Boundedness Property

According to the Boundedness Property, if \( m \) and \( M \) are the minimum and maximum values of a function \( f \) over an interval \([a, b]\), then the definite integral \( \int_a^b f(x) \, dx \) is bounded by \( m(b-a) \leq \int_a^b f(x) \, dx \leq M(b-a) \).

Step 4: Calculate the Bounds

Substitute the values of \( m \) and \( M \) from Step 2 and the interval \([0.2, 0.4]\) into the Boundedness Property formula to calculate the bounds. \( m = 0.002 \) and \( M = 0.0021 \), so: \( m(0.4 - 0.2) = 0.002 \times 0.2 = 0.0004 \) and \( M(0.4 - 0.2) = 0.0021 \times 0.2 = 0.00042 \).

Step 5: Conclude the Bounds

Based on the calculations, the lower bound for the integral is 0.0004 and the upper bound is 0.00042.

Key Concepts

Graphing Calculator

A graphing calculator is a powerful tool that helps visualize mathematical functions, including calculus concepts. By plotting the function on a graph, you can see how it behaves over a given interval.
For the exercise, graphing the function \( f(x) = 0.002 + 0.0001 \cos^2 x \) from 0.2 to 0.4 allows you to assess its behavior between these bounds. Understanding where the function rises or falls aids in identifying critical features like minima and maxima.
Using a graphing calculator, students can perform the following steps:

• Input the function equation into the calculator.
• Set the graph to display over the interval [0.2, 0.4].
• Examine the plotted function to determine approximate minimum and maximum values.

These steps align with an intuitive approach to analyzing functions visually, making graphing calculators invaluable companions in calculus.

Definite Integrals

Definite integrals calculate the accumulated total of a function over a specific interval. They provide the net area under the function's curve, which may be positive or negative depending on whether the function lies above or below the x-axis.
In this case, the integral \( \int_{0.2}^{0.4}(0.002 + 0.0001 \cos^2 x) \, dx \) helps find the area under the curve from 0.2 to 0.4. Calculus students apply these concepts to numerous real-world situations, like finding displacement or computing profits.
When evaluating definite integrals using functions like this one, remember:

• Review the limits of integration, which are your bounds.
• Understand that integration is about finding the total area within those bounds.
• Consider the boundedness of the function within the limits to estimate the integral's value.

Understanding definite integrals encourages a comprehensive grasp of how functions behave across specific intervals, vital to higher-level calculus applications.

Boundedness Property

The Boundedness Property is a useful theorem in calculus for approximating the limits of definite integrals.
It asserts that if a continuous function holds minimum and maximum values, \( m \) and \( M \), within a closed interval \([a, b]\), then the integral value is bounded between \( m(b-a) \leq \int_a^b f(x) \ dx \leq M(b-a) \).
In our example, first graphing the function helps ascertain these minimum and maximum values. For \( f(x) = 0.002 + 0.0001 \cos^2 x \), the reasoning is:

• Identify \( m \) and \( M \) from the graph.
• Calculate \( m(b-a) \) and \( M(b-a) \) with the interval lengths.

The computed bounds, in this case, demonstrate the integral's true value lies within these calculated bounds. The boundedness property simplifies evaluating integrals, especially when an exact numeric answer may not be readily apparent.

Trigonometric Functions

Trigonometric functions like cosine play significant roles in calculus due to their periodic nature and wide-ranging applications. With expressions like \( \cos^2 x \) within integrands, understanding these functions becomes crucial.
The function \( \cos x \) has a repetitive cycle from -1 to 1. Squaring \( \cos x \) in \( \cos^2 x \) confines the range from 0 to 1, ensuring non-negative values, crucial for integration. In this problem, the function changes minimally due to the small coefficient for \( \cos^2 x \).
Some essential properties of trigonometric functions include:

• Periodic nature, repeating every \(2\pi\).
• Utility in modeling real-world phenomena like sound waves.
• Behavioral changes when scalar multiples or powers are applied.

Developing a familiarity with trigonometric functions supports recognizing how they influence integrands, highlighting their far-reaching impact in mathematical and physical contexts.