Problem 52 - Calculus Early Transcendentals

Question

Use the approaches discussed in this section to evaluate the following integrals. $$\int_{0}^{\pi / 8} \sqrt{1-\cos 4 x} d x$$

Step-by-Step Solution

Answer

Question: Evaluate the integral \(\int_{0}^{\pi / 8} \sqrt{1-\cos 4 x}\, d x\) Answer: The integral evaluates to \(-\frac{1}{2} + \frac{\sqrt{2}}{2}\).

Step 1: Use half-angle formulas to simplify the integrand.

We have a cosine of 4x inside the square root, we can use the half-angle formula given by: $$\cos 2 \theta = 2 \cos^2 \theta - 1$$ Let's substitute 2x as our θ: $$\cos 4 x = 2 \cos^2 2 x - 1$$ Now, we can rewrite the integrand as: $$\sqrt{1 - \cos 4x} = \sqrt{1 - (2\cos^2 2x - 1)} = \sqrt{2 - 2\cos^2 2x}$$

Step 2: Change the integrand into sine function.

Using the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\), we can replace the \(\cos^2 2x\)... $$\sin^2 2x = 1 - \cos^2 2x$$ So the integrand becomes: $$\sqrt{2 - 2\cos^2 2x} = \sqrt{2 - 2(1-\sin^2 2x)} = \sqrt{2\sin^2 2x} = \sqrt{2}\sin 2x$$ Thus, the integral becomes: $$\int_{0}^{\pi / 8} \sqrt{1-\cos 4 x}\, d x = \int_{0}^{\pi / 8} \sqrt{2}\sin 2 x\, d x$$

Step 3: Integrate the simplified function.

Now we can integrate the simplified function: $$\int_{0}^{\pi / 8} \sqrt{2}\sin 2 x\, d x = \sqrt{2} \int_{0}^{\pi / 8} \sin 2x\, dx$$ Integration of \(\sin 2x\): $$\int \sin 2x\, dx = -\frac{1}{2}\cos 2x + C$$ Now, apply the definite integral limits: $$\sqrt{2} \left[ -\frac{1}{2}\cos 2x \right]_{0}^{\pi / 8} = \sqrt{2} \left( -\frac{1}{2}\cos\frac{\pi}{4} - \left(-\frac{1}{2}\cos 0 \right) \right)$$

Step 4: Evaluate the definite integral.

Now, we can substitute the limits and simplify: $$\sqrt{2} \left(-\frac{1}{2}\cos\frac{\pi}{4} + \frac{1}{2}\cos 0 \right) = \sqrt{2} \left(-\frac{1}{2}\frac{\sqrt{2}}{2} + \frac{1}{2}\cdot 1 \right)$$ This further simplifies to: $$=\sqrt{2} \left(-\frac{1}{4}\sqrt{2} + \frac{1}{2} \right) = -\frac{1}{2} + \frac{\sqrt{2}}{2}$$ So the definite integral result is: $$\int_{0}^{\pi / 8} \sqrt{1-\cos 4 x}\, d x = -\frac{1}{2} + \frac{\sqrt{2}}{2}$$

Key Concepts

Half-Angle Formulas

Half-angle formulas are essential tools in integral calculus, especially when dealing with trigonometric functions. A common half-angle identity is the expression for cosine in terms of double angles:

• For cosine, it is expressed as: \( \cos 2\theta = 2\cos^2 \theta - 1 \)
• By solving for \(\cos^2 \theta\), we get: \( \cos^2 \theta = \frac{1 + \cos 2\theta}{2} \)

In our specific problem, we have \( \cos 4x \), which corresponds to a double angle 2x where \( \theta = 2x \). By substituting \( 2x \) into the half-angle formula, \( \cos 4x = 2\cos^2 2x - 1 \) becomes useful to simplify inside the integral. This technique changes trigonometric integrals into simpler forms for straightforward calculation. Understanding and applying these formulas can significantly reduce calculation difficulty in integrals involving trigonometric expressions.

Definite Integrals

Definite integrals are a fundamental concept in integral calculus, providing a way to calculate the net area under a curve between two specified limits. These bounds are set values, like from 0 to \(\pi/8\) in our exercise. When evaluating a definite integral, follow these steps:

• Resolve the integrand: Simplify the function using identities or substitutions, as we've seen with trigonometric identities.
• Calculate the integral: Apply standard integral forms and rules to find the antiderivative.
• Apply limits: Substitute the upper bound and lower bound into the antiderivative, then compute their difference.

In our scenario, we first convert \( \sqrt{1-\cos 4x} \) to a simpler form using trigonometric identities. Then we performed the integration followed by substituting the limits \(0\) and \(\pi/8\) in the antiderivative. Evaluating definite integrals offers precise numerical results and requires a good understanding of integration techniques and properties.

Trigonometric Identities

Trigonometric identities are equations that are true for all angles and are particularly usable in simplifying calculus problems.

• Pythagorean Identity: This identity \( \sin^2 x + \cos^2 x = 1 \) helps to switch between sine and cosine expressions. It's a cornerstone in transforming expressions.
• Double Angle Formulas: These, like \( \cos 2x = 2\cos^2 x - 1 \), balance larger angle expressions and connect them to standard functions, reducing complexities in integrals.
• Other Useful Identities: From angle sum formulas to half-angle transformations, they streamline solvers' work in calculus problems.

In our integral calculation, transforming \( \cos^2 2x \) to \( 1 - \sin^2 2x \) by using the Pythagorean identity was crucial. This step simplified what could be a complex integration into something manageable. Recognizing and applying such identities can drastically alter the difficulty level of solving integrals.