Problem 8 - Thomas Calculus

Question

Evaluate the integrals in Exercises \(1-14\) $$ \int_{0}^{1} 8 \cos ^{4} 2 \pi x d x $$

Step-by-Step Solution

Answer

The value of the integral is 3.

Step 1: Identify the Integral

The given integral is \( \int_{0}^{1} 8 \cos ^{4} (2 \pi x) \ dx \). The goal is to evaluate this definite integral.

Step 2: Use a Trigonometric Identity

Simplify \( \cos^4(2\pi x)\) using the identity: \( \cos^2(\theta) = \frac{1 + \cos(2\theta)}{2} \). We can apply this twice.First, \( \cos^2(2\pi x) = \frac{1 + \cos(4\pi x)}{2} \).Then, \(\cos^4(2\pi x) = \left(\frac{1 + \cos(4\pi x)}{2}\right)^2 = \frac{(1 + \cos(4\pi x))^2}{4}.\)

Step 3: Expand the Squared Term

Expand \((1 + \cos(4\pi x))^2\) to simplify further: \[(1 + \cos(4\pi x))^2 = 1 + 2\cos(4\pi x) + \cos^2(4\pi x).\]

Step 4: Simplify Further Using Identities

Replace \( \cos^2(4\pi x) \) using the identity again:\[\cos^2(4\pi x) = \frac{1 + \cos(8\pi x)}{2}.\]Thus,\[1 + 2\cos(4\pi x) + \cos^2(4\pi x) = 1 + 2\cos(4\pi x) + \frac{1 + \cos(8\pi x)}{2}.\]

Step 5: Integrate Each Term

Substitute back into the integral and calculate:\[\int_{0}^{1} 8 \cdot \frac{1 + 2\cos(4\pi x) + \frac{1 + \cos(8\pi x)}{2}}{4} \ dx.\]This simplifies to:\[\int_{0}^{1} 2 \left(\frac{3}{2} + 2\cos(4\pi x) + \frac{1}{2}\cos(8\pi x) \right) \ dx.\]Integrate term by term over the interval \([0,1]\).

Step 6: Evaluate the Simplified Integral

The integral simplifies to:\[2 \cdot \left( \frac{3}{2}\int_{0}^{1} 1 \ dx + 2\int_{0}^{1} \cos(4\pi x) \ dx + \frac{1}{2}\int_{0}^{1} \cos(8\pi x) \ dx \right).\]Evaluate these:- \( \int_{0}^{1} 1 \ dx = 1. \)- \( \int_{0}^{1} \cos(4\pi x) \ dx = 0 \text{ (since it's a complete period of cosine)}. \)- \( \int_{0}^{1} \cos(8\pi x) \ dx = 0 \text{ (also complete period).} \)

Step 7: Compute the Final Result

Using the evaluated integrals, we get:\[2 \cdot \left( \frac{3}{2} \cdot 1 + 2 \cdot 0 + \frac{1}{2} \cdot 0 \right) = 2 \cdot \frac{3}{2} = 3.\]

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that are true for all values of the variable where the functions are defined. They play a crucial role in simplifying expressions, especially in integral calculus. In this exercise, the identity used is \( \cos^2(\theta) = \frac{1 + \cos(2\theta)}{2} \). This identity helps transform powers of cosine into simpler expressions. By reducing powers of trigonometric functions, calculations become much more manageable.

In our problem, first \( \cos^2(2\pi x) \) is simplified using the identity, and then \( \cos^4(2\pi x) \) is expressed as \( \left(\frac{1 + \cos(4\pi x)}{2} \right)^2 \). After expanding and further simplifying using \( \cos^2(4\pi x) \), we make integral evaluation possible.

This method illustrates how integral calculus often relies on these identities to transform complex trigonometric expressions into integrable functions.

Integral Calculus

Integral calculus is a branch of mathematics that deals with the concept of an integral. It allows us to calculate quantities based on rates of change. In essence, integrals calculate the area under a curve. When faced with definite integrals, we evaluate these areas over specified bounds.

The definite integral in this problem is \( \int_{0}^{1} 8 \cos^4(2 \pi x) \ dx \). The main goal is to find the exact area under the curve of the function \( 8 \cos^4(2 \pi x) \) from \(x = 0\) to \(x = 1\). By applying trigonometric identities, the function simplifies, allowing us to decompose it into manageable parts.

Integral calculus involves breaking down the problem - simplifying the function via mathematical manipulation and then dealing with simpler terms. Understanding these concepts ensures one can approach and solve integrals in an efficient manner.

Evaluation of Integrals

The evaluation of integrals involves calculating the net area under a curve defined by a function, within given bounds. This requires integrating the simplified expression and substituting the definite limits to find a specific numerical value.

In this problem, once the integral \( \int_{0}^{1} 8 \cos^4(2 \pi x) \ dx \) is simplified to involve trigonometric components, each part is individually integrated. For functions like \( \cos(kx) \), where \(k\) is a constant, integration over a complete period results in zero, as was the case for both \( \int_{0}^{1} \cos(4 \pi x) \ dx \) and \( \int_{0}^{1} \cos(8 \pi x) \ dx \).

This leads to the final computed result, demonstrating how integral evaluation combines trigonometric simplification, correct use of integration rules, and proper limit substitution to find the answer. Mastery of these steps is crucial for solving problems in integral calculus, ensuring accurate and effective results.