Problem 44 - Calculus: Early Transcendental Functions

Question

Find the given area. The area between \(y=\sin x \quad\) and the \(\quad x\) -axis for \(-\pi / 2 \leq x \leq \pi / 4\)

Step-by-Step Solution

Answer

The total area between the curve \(y = \sin x\) and the x-axis from \(-\pi/2\) to \(\pi/4\) can be obtained by calculating the separate areas for negative and positive \(x\) values, then adding them together: \(Area = -\int_{-\pi/2}^{0} \sin x \, dx + \int_{0}^{\pi/4} \sin x \, dx \)

Step 1: Analyze the direction

The function \(y = \sin x\) is positive when \(x\) is between 0 and \(\pi/2\), and negative when \(x\) is between \(-\pi/2\) and 0. Thus, we'll analyze the function separately for these two ranges.

Step 2: Calculate the area for negative x values

For x values between \(-\pi/2\) and 0, we need to negate the area because the curve is below the x-axis. Thus, we calculate the area as: \(-\int_{-\pi/2}^{0} \sin x \, dx\)

Step 3: Calculate the area for positive x values

For x values between 0 and \(\pi/4\), the curve is above the x-axis. So, we calculate the area as: \(\int_{0}^{\pi/4} \sin x \, dx\)

Step 4: Calculate the total area

Add the absolute values of the areas calculated in Step 2 and Step 3 to find the total area between the curve and the x-axis between \(-\pi/2\) and \(\pi/4\).

Key Concepts

Integration

Integration is an essential concept in calculus that allows us to find the accumulated quantity, such as area under a curve. It can be understood as the reverse process of differentiation and primarily focuses on summing an infinite number of infinitesimally small quantities. When we integrate a function, we are essentially calculating the net area between that function and the horizontal axis over a specific interval.

There are two types of integrals:

• Indefinite Integral: Integration without boundaries, representing a family of functions and including an arbitrary constant, often denoted by "C". Represented as \(\int f(x) \, dx\).
• Definite Integral: Integration within specific bounds, viz., it computes the exact area under a curve between two points on the x-axis. Denoted as \[ \int_{a}^{b} f(x) \, dx \], this is the focus in our exercise.

Integration, especially definite integrals, is a powerful tool for understanding and solving problems involving areas, volumes, and other cumulative metrics.

Definite Integral

A definite integral assigns a numerical value representing the signed area between the curve of a function and the x-axis over a specific interval. For instance, in our exercise, we seek the area between the curve \(y = \sin x\) and the x-axis from \(-\pi/2\) to \(\pi/4\).

Here's how it works:

• The lower limit \(a\) and upper limit \(b\) define the range along the x-axis.
• The definite integral is calculated by finding the antiderivative of the function and evaluating it at these two boundaries. The result is computed as the difference: \[ F(b) - F(a) \], where \( F \) is the antiderivative of \( f \).

In solving this problem, we separate the given range into two parts based on where the curve is above and below the x-axis because the sine function goes negative and positive depending on the x-values. It's crucial to account for these changes to accurately find the total area, necessitating us to take the absolute value of each integral before summing them up.

Trigonometric Functions

Trigonometric functions are vital mathematical functions such as \( \sin(x) \), \( \cos(x) \), and \( \tan(x) \), which relate angles of a triangle to the lengths of its sides. In calculus, they also help describe wave-like patterns and periodic behaviors.

The sine function, \( \sin(x) \), specifically portrays a wave oscillating between 1 and -1 over the interval \([0, 2\pi]\). For the problem provided, \( y = \sin x \) holds particular characteristics:

• \( \sin x \) is positive in the first quadrant (from 0 to \( \pi/2\)) and negative in the fourth quadrant (from \( -\pi/2\) to 0).
• This alternating nature between positive and negative parts necessitates additional consideration when calculating areas using definite integrals.

Understanding the fundamental behavior of trigonometric functions like sine is crucial for solving problems involving integrals, as it guides how to approach splitting the function's integration depending on its position relative to the x-axis.