Problem 63 - Thomas Calculus

Question

Evaluate each integral in Exercises \(63-70\) by eliminating the square root. $$ \int_{0}^{2 \pi} \sqrt{\frac{1-\cos x}{2}} d x $$

Step-by-Step Solution

Answer

The value of the integral is 4.

Step 1: Identify the Trigonometric Identity

Notice that the expression under the square root, \( \frac{1 - \cos x}{2} \), resembles a known trigonometric identity. This identity is, \(\sin^2(x/2) = \frac{1 - \cos x}{2}\). This allows us to rewrite the square root as \( \sqrt{\frac{1 - \cos x}{2}} = \sin(x/2) \).

Step 2: Rewrite the Integral

Substitute the expression we found in Step 1 into the integral. The integral becomes: \[ \int_{0}^{2\pi} \sin(x/2) \; dx \]

Step 3: Make a Substitution

To simplify the integral, use the substitution \(u = x/2\). Therefore, \(du = (1/2)dx\) or equivalently, \(dx = 2du\). The limits of integration change as well: when \(x=0\), \(u=0\); and when \(x=2\pi\), \(u=\pi\). Substitute these into the integral: \[ \int_{0}^{\pi} \sin(u) \cdot 2 \; du \]

Step 4: Evaluate the Integral

Integrate \(2 \sin(u)\) with respect to \(u\): \[ 2 \int \sin(u) \; du = -2 \cos(u) + C \] Now apply the limits of integration: \[ [-2 \cos(u)]_{0}^{\pi} = -2\cos(\pi) - (-2\cos(0)) \]

Step 5: Compute the Definite Integral

Substitute the values for the cosine terms: \[ -2(-1) - (-2(1)) = 2 + 2 = 4 \]

Step 6: Final Solution

Double-check the computations to confirm the calculations: The integral from 0 to \(2\pi\) of \( \sqrt{\frac{1 - \cos x}{2}} \; dx \) simplifies to 4.

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental tools in calculus that help simplify complex expressions involving trigonometric functions. These identities are essentially equations that hold true for any value of the involved variables. In this topic, our focus is on a particular identity:

• \( rac{1 - \cos x}{2} = \sin^2(x/2) \)

This identity is crucial in the given problem. It allows us to transform the expression under the square root, simplifying it from \( \sqrt{\frac{1 - \cos x}{2}} \) into a more manageable form: \( \sin(x/2) \). This substitution not only eliminates the square root but also makes the expression easier to integrate later on.
Recognizing and applying trigonometric identities is a powerful technique to resolve otherwise difficult integral calculations, as it transitions a problem to a simpler expression.

Definite Integral

A definite integral represents the area under a curve within a specified interval. In this situation, we seek the integral of \( \sin(x/2) \) from 0 to \( 2\pi \).
The process of calculating a definite integral involves these key steps:

• Evaluate the antiderivative of the function, which in this case is the function we integrate over the variable \(x\).
• Apply the limits of integration by substituting the upper limit into the antiderivative and then subtracting the value of the antiderivative evaluated at the lower limit.

For our problem, after applying the integration techniques, we determine the area using the antiderivative \(-2\cos(u)\) evaluated from 0 to \(\pi\). Understanding definite integrals also provides insight into the geometry of a problem, as they often represent physical quantities like area or displacement.

Substitution Method

The substitution method in calculus is a technique used to simplify the process of evaluating integrals, by transforming them into an easier form. It's akin to reversing the chain rule for derivatives. Here's how it works:

• Identify a substitution that simplifies the integral. In the problem, this was \(u = x/2\), transforming the integral from being with respect to \(x\) to \(u\).
• Compute the differential of the new variable \(u\), giving us \(du = (1/2) dx\), which implies \(dx = 2 du\).
• Replace all instances of the original variable and its differential in the integral, including changing the limits of integration.

Implementing these steps converts the integral \( \int_{0}^{2\pi} \sin(x/2) \; dx \) to \( 2 \int_{0}^{\pi} \sin u \; du \), streamlining the calculation. Essentially, the substitution method is a strategic approach to "reframe" an integral to a simpler format, making it more manageable to solve.