Problem 60 - Calculus Early Transcendentals

Question

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

Step-by-Step Solution

Answer

Question: Evaluate the definite integral $$\int_{0}^{\pi/2} \sin^4{\theta} d\theta.$$ Answer: The definite integral evaluates to $$\int_{0}^{\pi / 2} \sin^{4} \theta d \theta = \frac{3\pi}{16}.$$

Step 1: Use power reduction formula for sine

First, apply the power reduction formula to the integrand: $$\sin^4{\theta} = (\sin^2{\theta})^2 = \left(\frac{1 - \cos(2\theta)}{2}\right)^2.$$ Now, expand and simplify the expression.

Step 2: Expand and simplify the expression

Expand the squared expression of the integrand: $$\left(\frac{1 - \cos(2\theta)}{2}\right)^2 = \frac{1 - 2\cos(2\theta) + \cos^2(2\theta)}{4}.$$ Apply the power reduction formula for cosine to the term \(\cos^2(2\theta)\): $$\cos^2(2\theta) = \frac{1 + \cos(4\theta)}{2}.$$ Substitute back into the expanded expression: $$\frac{1 - 2\cos(2\theta) + \cos^2(2\theta)}{4} = \frac{1 - 2\cos(2\theta) + \frac{1 + \cos(4\theta)}{2}}{4} = \frac{3 - 4\cos(2\theta) + \cos(4\theta)}{8}.$$

Step 3: Substitute the simplified integrand into the integral and calculate the integral

Now, substitute the simplified expression back into the integral and evaluate it: $$\int_{0}^{\pi/2} \sin^4{\theta} d\theta = \int_{0}^{\pi/2} \frac{3 - 4\cos(2\theta) + \cos(4\theta)}{8} d\theta.$$ Integrate each term separately: $$\int_{0}^{\pi/2} \frac{3}{8} d\theta - \int_{0}^{\pi/2} \frac{4\cos(2\theta)}{8} d\theta + \int_{0}^{\pi/2} \frac{\cos(4\theta)}{8} d\theta.$$

Step 4: Evaluate each term in the integral

Evaluate each term in the separated integral: $$\left[\frac{3\theta}{8}\right]_{0}^{\pi/2} - \left[\frac{2\sin(2\theta)}{8}\right]_{0}^{\pi/2} + \left[\frac{\sin(4\theta)}{32}\right]_{0}^{\pi/2}.$$ $$\left[\frac{3\pi}{16} - \frac{2\sin(\pi)}{8} + \frac{\sin(2\pi)}{32}\right] - \left[\frac{3(0)}{8} - \frac{2\sin(0)}{8} + \frac{\sin(0)}{32}\right] = \frac{3\pi}{16}.$$ So the integral evaluates to: $$\int_{0}^{\pi / 2} \sin ^{4} \theta d \theta = \frac{3\pi}{16}.$$

Key Concepts

Power Reduction Formula

The power reduction formula is a handy tool when integrating trigonometric functions that are raised to a power higher than one. This formula helps in simplifying expressions by reducing the power of trigonometric functions.

For sine and cosine functions, the power reduction formulas are:

• \( ext{For } \sin^2 \theta: \sin^2 \theta = \frac{1 - \cos(2\theta)}{2} \)
• \( ext{For } \cos^2 \theta: \cos^2 \theta = \frac{1 + \cos(2\theta)}{2} \)

When dealing with higher powers, like \( \sin^4 \theta \), we first express it as \( (\sin^2 \theta)^2 \).

Then, we apply the power reduction formula for \(\sin^2 \theta\). This step is essential as it transforms a complex integral into a simpler one. Now, the integrand involves cosine functions, which are often easier to integrate. Ultimately, the power reduction formula is invaluable in elegantly solving higher power trigonometric integrals.

Trigonometric Integrals

Integrals involving trigonometric functions such as \( \sin \theta\), \( \cos \theta\), and others are referred to as trigonometric integrals. These integrals are a major area of study in calculus and require specific techniques to evaluate.

When dealing with integrals like \( \int \sin^n \theta \, d\theta \), where the exponent \( n \) is greater than 1, strategies such as substitution, using identities, or applying power reduction formulas come into play. Calculating these integrals often involves familiar transformations and substitutions that simplify the trigonometric function's expression.

For example, using trigonometric identities, we transform \( \sin^4 \theta \) into simpler terms, involving just \( \cos \) and constants, making the integral solvable with basic integration techniques. Understanding these transformations simplifies solving complex trigonometric integrals and is vital for evaluating definite integrals over specified limits.

Integration Techniques

Integration techniques are essential tools in calculus that help in finding antiderivatives, and they are especially valuable when direct integration isn't straightforward.

For integrals presenting trigonometric functions raised to a power, an effective method is to break down the integral using known identities. After transforming the integral using these identities, we can apply standard integration rules.

• Split the integral into simpler parts, as seen in the step where we break down \( \left(\frac{3}{8} - \frac{4\cos(2\theta)}{8} + \frac{\cos(4\theta)}{8}\right) \)
• Integrate each term separately and independently to manage the process.
• Evaluate each term over the specified limits to find the definite integral result.

Using these techniques, integrates seemingly difficult trigonometric expressions smoothly into simpler antiderivatives, leading to exact results when limits are applied. Mastery of these techniques is crucial for efficient and successful integration across various calculus problems.