Problem 49 - Calculus Early Transcendentals: Pearson New International Edition

Question

If \(f\) is periodic with period \(p\), then $$ \int_{a}^{a+p} f(x) d x=\int_{0}^{p} f(x) d x $$ Convince yourself that this is true by drawing a picture and then use the result to calculate \(\int_{1}^{1+\pi}|\sin x| d x\).

Step-by-Step Solution

Answer

The integral is 2, due to symmetry and the periodicity of \( |\sin(x)| \).

Step 1: Understanding Periodicity

A periodic function with period \(p\) means that \(f(x) = f(x+p)\) for all \(x\). This implies that the integral over one period is the same regardless of the starting point.

Step 2: Apply Periodicity Property to Integrals

Given that \(f\) is periodic with period \(p\), we have \(\int_a^{a+p} f(x) \, dx = \int_0^p f(x) \, dx\). This holds because shifting the interval by the period \(p\) does not change the area under one complete cycle of the function.

Step 3: Visualize the Function

Drawing a picture of the function \(f(x) = |\sin(x)|\), notice that it repeats every \(\pi\) units, indicating a period of \(\pi\) for \(|\sin(x)|\). This can help reinforce that the integral over a full cycle \(a\) to \(a+p\) is the same as from \(0\) to \(p\).

Step 4: Set up the Integral with Given Limits

Since the period \(p\) of \(|\sin(x)|\) is \(\pi\), we apply the periodicity property: \(\int_1^{1+\pi} |\sin(x)| \, dx = \int_0^\pi |\sin(x)| \, dx\).

Step 5: Calculate the Integral for One Period

The integral \(\int_0^\pi |\sin(x)| \, dx\) can be evaluated since \(|\sin(x)|\) over one period is equivalent to the function \(\sin(x)\) because \(\sin(x)\) is non-negative in this interval. Thus, \(\int_0^\pi |\sin(x)| \, dx = 2\), using symmetry or direct integral calculation.

Key Concepts

Integral Calculus

Integral Calculus is a fundamental branch of mathematics that studies the process of integration.It helps in finding the total size or value, such as areas under curves or the accumulated quantity over an interval.When working with periodic functions, integration simplifies thanks to the properties these functions exhibit.
For instance, if a function is periodic with period \( p \), the integral over one full period is the same regardless of the starting point.This is incredibly useful for functions like \( |\sin(x)| \), where performing the integral from \( a \) to \( a + p \) provides the same result as from \( 0 \) to \( p \).
Integrals help in:

• Calculating the area beneath curves
• Finding accumulated quantities
• Solving for distance, area, and volume in physical applications

Using this principle, we can take a complicated integral and transform it into a simpler one by using periodicity.

Absolute Value Function

The absolute value function, denoted as \( |x| \), is a pivotal concept in mathematics, representing the distance of a number from zero without considering direction.For example, \(|3|\) and \(|-3|\) both equal 3, signifying they are both 3 units away from zero along the number line.This concept is crucial when dealing with functions that include \(|\sin(x)|\).
This function is particularly important:

• It ensures all function outputs are non-negative.
• It affects periodic functions by making the range always positive.
• It allows you to handle scenarios where direction or sign should be ignored.

In problems related to periodic functions like \(|\sin(x)|\), this concept ensures the resulting function, despite the oscillation, never drops below zero.

Properties of Sine Function

The sine function, \( \sin(x) \), is one of the core trigonometric functions characterized by its periodic nature.Periodicity implies that \( \sin(x + p) = \sin(x) \) for all \(x\), and for \(\sin(x)\), this period is \(2\pi\).However, when we consider the function \(|\sin(x)|\), its properties slightly shift due to the absolute value.
Key properties to know:

• The sine function oscillates between -1 and 1.
• \(|\sin(x)|\) oscillates between 0 and 1, removing negative values.
• \( |\sin(x)| \) is periodic with a period of \( \pi \).

The positive sine curve is repeated every \(\pi\) units due to the absolute value, which is useful when evaluating integrals over specific intervals that align with this period.Understanding these properties aids in interpreting and solving integral problems involving periodic functions.