Problem 50 - Calculus Early Transcendentals: Pearson New International Edition

Question

, use the Substitution Rule for Definite Integrals to evaluate each definite integral. $$ \int_{0}^{1 / 2} \sin (2 \pi x) d x $$

Step-by-Step Solution

Answer

The integral evaluates to \( \frac{1}{\pi} \).

Step 1: Identify the Substitution

We need to find an appropriate substitution that will simplify the integral. Here, let's consider the substitution \( u = 2\pi x \). This substitution is chosen because it matches the argument of the sine function, making the integral easier to solve.

Step 2: Find the Differential

Calculate the differential \( du \) from the substitution. Differentiating \( u = 2\pi x \) with respect to \( x \), we get \( \frac{du}{dx} = 2\pi \), so \( du = 2\pi dx \). Rearranging gives \( dx = \frac{du}{2\pi} \).

Step 3: Change the Limits of Integration

Substitute the limits of integration based on \( u = 2\pi x \). When \( x = 0 \), \( u = 2\pi(0) = 0 \). When \( x = \frac{1}{2} \), \( u = 2\pi\left(\frac{1}{2}\right) = \pi \). The new limits are from 0 to \( \pi \).

Step 4: Rewrite the Integral in Terms of \( u \)

Substitute everything into the integral: \[ \int_{0}^{\pi} \sin(u) \cdot \frac{du}{2\pi} \]. This simplifies the problem, making it easier to integrate.

Step 5: Integrate the Function

The integral of \( \sin(u) \) is \( -\cos(u) \). Thus, the integral becomes \( \int_{0}^{\pi} \sin(u) \cdot \frac{du}{2\pi} = -\frac{1}{2\pi} \left[ \cos(u) \right]_{0}^{\pi} \).

Step 6: Evaluate the Definite Integral

Evaluate \(-\frac{1}{2\pi} \left[ \cos(u) \right]_{0}^{\pi} \). Substituting the bounds, we have: \(-\frac{1}{2\pi} \left( \cos(\pi) - \cos(0) \right) = -\frac{1}{2\pi} (-1 - 1) \). This simplifies to \( \frac{1}{\pi} \).

Key Concepts

Substitution Rule

The substitution rule is a crucial integration technique that simplifies the evaluation of integrals. This method involves transforming an integral into a more manageable form by using a substitution that changes the variable of integration.
The goal is to choose a substitution that makes the integrand, the function being integrated, easier to integrate. In the context of our problem, the substitution helps convert the trigonometric integral into a simpler form.

• Identify the substitution: Choose a function of the variable that appears in the integrand; for this problem, it's best to select something that reduces complexity, like matching the argument inside a trigonometric function.
• Calculate the differential: This involves finding the derivative of your substitution function with respect to the original variable.
• Recalculate the integral limits: Substitute your new variable into the original limits to determine the new bounds.

The substitution rule essentially changes the perspective of the problem, helping to solve potentially complex integrals more easily by simplifying the functions you're working with.

Definite Integrals

Definite integrals calculate the net area between a function and the x-axis over a specific interval. Unlike indefinite integrals, which represent a family of functions, definite integrals result in a number.
To solve definite integrals, it's crucial to find the antiderivative of the function and evaluate it at the upper and lower limits of integration.

• In our exercise, we used the antiderivative to transform the function from its integrated form back to a simple computation.
• The Fundamental Theorem of Calculus connects differentiation and integration, allowing for the evaluation of definite integrals through an antiderivative.

Definite integrals are incredibly useful in physics and engineering because they easily quantify accumulated quantities such as area, volume, and even probabilities. They always have specific real-world applications and provide precise values which reflect changes over specified intervals.

Trigonometric Integrals

Trigonometric integrals involve integrating expressions that include trigonometric functions like sine, cosine, tangent, etc. These types of integrals appear frequently in calculus due to the periodic nature of trig functions.
This exercise specifically involves integrating a sine function.

• Key strategy: When integrating products or sums of trig functions, look for identities or substitutions that can simplify the expression.
• The integral of basic trigonometric functions is often remembered from integral tables, for instance, the integral of \( \sin(u) \) is \( -\cos(u) \).

Trigonometric integrals require an understanding of trigonometric identities and their relationships. Often, adopting substitutions to simplify expressions can reveal hidden patterns or facilitate the cancellation of terms, as illustrated in this exercise by using the substitution rule. Trigonometric integrals provide a gateway into more complex analytical problems, building a foundation for solving real-world trigonometric applications.