Problem 56 - Calculus Early Transcendentals

Question

Evaluate the following integrals. $$\int_{0}^{\pi / 4} \cos ^{2} 8 \theta d \theta$$

Step-by-Step Solution

Answer

Question: Evaluate the definite integral $$\int_{0}^{\pi/4} \cos^2 (8\theta) \, d\theta$$. Answer: The value of the integral is $$\frac{\pi}{8}$$.

Step 1: Apply the double angle formula for cosine

We will use the double angle formula for cosine to simplify the integrand. The double angle formula for cosine states that $$\cos(2\alpha) = 2\cos^2(\alpha) - 1.$$ Using this formula, we can rewrite the integrand as follows: $$\cos^2(8\theta) = \dfrac{1+\cos(16\theta)}{2}.$$ Now, our integral becomes: $$\int_{0}^{\pi / 4} \cos^2 (8\theta) \, d\theta = \frac{1}{2}\int_{0}^{\pi/4} (1+\cos(16\theta)) \, d\theta.$$

Step 2: Evaluate the integral of the simplified expression

We can now separate this integral into the sum of two simpler integrals: $$\frac{1}{2}\int_{0}^{\pi/4} (1+\cos(16\theta)) \, d\theta = \frac{1}{2} \left(\int_{0}^{\pi/4} 1 \, d\theta + \int_{0}^{\pi/4} \cos(16\theta) \, d\theta \right).$$ Evaluate each of the integrals separately: $$\int_{0}^{\pi/4} 1 \, d\theta = \theta\Big|_{0}^{\pi/4} = \frac{\pi}{4} - 0 = \frac{\pi}{4}.$$ $$\int_{0}^{\pi / 4} \cos(16\theta) \, d\theta = \frac{1}{16}\sin(16\theta)\Big|_{0}^{\pi / 4} = \frac{1}{16}(\sin(4\pi)-\sin(0)) = \frac{1}{16}(0-0) = 0.$$ Now that we have the values for the individual integrals, let's plug them back into the full integral: $$\frac{1}{2} \left(\int_{0}^{\pi/4} 1 \, d\theta + \int_{0}^{\pi/4} \cos(16\theta) \, d\theta \right) = \frac{1}{2} \left(\frac{\pi}{4} + 0\right) = \frac{\pi}{8}.$$

Step 3: State the final result

Therefore, the value of the integral $$\int_{0}^{\pi/4} \cos^2(8\theta) \, d\theta$$ is $$\frac{\pi}{8}$$.

Key Concepts

Double Angle Formula

One of the most powerful tools in trigonometry is the double angle formula, especially when it comes to integrating trigonometric functions. The double angle formula for cosine is given by:
\[\cos(2\alpha) = 2\cos^2(\alpha) - 1.\]
This formula lets us express a squared cosine function in terms of a single cosine function with twice the angle, significantly simplifying the integration process. In our textbook problem, we used it to transform \(\cos^2(8\theta)\) into a more manageable form:
\[\cos^2(8\theta) = \frac{1+\cos(16\theta)}{2}.\]
By converting \(\cos^2\) to an expression with \(\cos\) and a constant, we could easily integrate it over the given interval. Understanding and applying this formula is critical when dealing with integrals involving trigonometric functions.

Cosine Function

Properties and Integration of Cosine

The cosine function is one of the fundamental trigonometric functions. The function \(\cos(\theta)\) represents the x-coordinate of a point on the unit circle at an angle \(\theta\) from the positive x-axis. The cosine function has several important properties that are useful in calculus:

• It is an even function, meaning \(\cos(-\theta) = \cos(\theta)\).
• It has a period of \(2\pi\), so it repeats every \(2\pi\) radians.
• The cosine of a sum of two angles can be expressed in terms of the sines and cosines of the individual angles.

When integrating the cosine function, knowing its antiderivative is essential. The integral of \(\cos(\theta)\) with respect to \(\theta\) is \(\sin(\theta)\), which we used in the solution of our exercise in evaluating the integral of \(\cos(16\theta)\).

Definite Integral

Computing Area Under the Curve

The definite integral is a fundamental concept in calculus that represents the net area under a curve over a certain interval. It's given by:
\[\int_{a}^{b} f(x) \, dx,\]
where \(a\) and \(b\) are the bounds of the interval, and \(f(x)\) is the function to be integrated. To calculate a definite integral, we find the antiderivative of \(f(x)\), evaluate it at the upper and lower bounds, and subtract the two values (commonly referred to as the Fundamental Theorem of Calculus). In the context of our exercise, the definite integral of the constant 1 on the interval \([0, \frac{\pi}{4}]\) yielded \(\frac{\pi}{4}\), and the integral involving the cosine function evaluated to 0. As a key concept, mastering definite integrals allows us to solve a plethora of problems in mathematics and applied sciences.