Problem 31 - University Calculus: Early Transcendentals

Question

Evaluate the integrals. $$\int_{0}^{1 / 2} \frac{4}{\sqrt{1-x^{2}}} d x$$

Step-by-Step Solution

Answer

The integral evaluates to \( \frac{2\pi}{3} \).

Step 1: Recognize the Integral Form

The integral \( \int \frac{4}{\sqrt{1-x^2}} \, dx \) is in a form that can be solved using a standard trigonometric substitution. The integrand suggests a connection to the derivative of the arcsine function.

Step 2: Apply Substitution

Let \( x = \sin(\theta) \). This implies \( dx = \cos(\theta) \, d\theta \). Also, \( \sqrt{1 - x^2} = \sqrt{1 - \sin^2(\theta)} = \cos(\theta) \).

Step 3: Change Limits of Integration

When \( x = 0 \), \( \sin(\theta) = 0 \) which means \( \theta = 0 \). When \( x = \frac{1}{2} \), \( \sin(\theta) = \frac{1}{2} \) which means \( \theta = \frac{\pi}{6} \). Therefore, the limits change from \([0, \frac{1}{2}]\) for \( x \) to \([0, \frac{\pi}{6}]\) for \( \theta \).

Step 4: Substitute and Simplify the Integral

Substitute \( x = \sin(\theta) \) into the integral: \( \int_0^{\pi/6} \frac{4}{\cos(\theta)} \cos(\theta) \, d\theta = \int_0^{\pi/6} 4 \, d\theta \).

Step 5: Evaluate the Integral

The integral now becomes \( \int_0^{\pi/6} 4 \, d\theta = 4\theta \Big|_0^{\pi/6} = 4 \left( \frac{\pi}{6} \right) - 4(0) = \frac{4\pi}{6} = \frac{2\pi}{3} \).

Key Concepts

Trigonometric Substitution

In calculus, trigonometric substitution is a powerful technique used to simplify integrals that involve square roots in the form of \( \sqrt{1-x^2} \), \( \sqrt{a^2-x^2} \), or \( \sqrt{x^2-a^2} \). The idea is to take advantage of trigonometric identities to rewrite the integral in a form that is easier to solve. For the integral \( \int \frac{4}{\sqrt{1-x^2}} \, dx \), notice that it aligns with the properties of \( \sin \theta \) because \( 1-\sin^2(\theta) = \cos^2(\theta) \).

• We substitute \( x = \sin(\theta) \), which leads to \( dx = \cos(\theta) \, d\theta \).
• This substitution transforms the expression under the square root: \( \sqrt{1-x^2} = \cos(\theta) \).

By applying trigonometric substitution, the original complex integral can be transformed into an expression that involves basic trigonometric functions. This greatly simplifies the integral, making it easier to evaluate and understand.

Definite Integrals

A definite integral represents the area under a curve from one point to another, within specified limits. For the integral expression \( \int_{0}^{1/2} \frac{4}{\sqrt{1-x^2}} \, dx \), these limits are from \( x = 0 \) to \( x = \frac{1}{2} \).

• Initially, the limits correspond to the variable \( x \).
• Once we apply the substitution \( x = \sin(\theta) \), the limits need to reflect \( \theta \) instead of \( x \).
• For our problem, when \( x = 0 \), \( \theta = 0 \), and when \( x = \frac{1}{2} \), \( \theta = \frac{\pi}{6} \).

Changing the limits in this way ensures that the area computed is based on the new trigonometric variable, reflecting the true regions on the graph where the change was applied. The evaluation of a definite integral not only provides a number that corresponds to this area but also allows us to use these properties in practical problems, such as finding distances or volumes.

Integral Evaluation Steps

To evaluate an integral effectively, especially when using substitution, it's important to follow each step methodically. Let's walk through the steps using trigonometric substitution as in our example.

• Recognize the form: Identify the integral type to determine the appropriate substitution method. For \( \int \frac{4}{\sqrt{1-x^2}} \, dx \), recognize it involves \( \sqrt{1-x^2} \).
• Apply substitution: Here, the substitution \( x = \sin(\theta) \) is applied, changing \( dx \) to \( \cos(\theta) \, d\theta \) and simplifying \( \sqrt{1-x^2} \) to \( \cos(\theta) \).
• Adjust integration limits: Convert the limits of \( x \) into limits for \( \theta \) based on the substitution.
• Simplify the integral: Substitute back into the integral and simplify. After substitution, it becomes \( \int_0^{\pi/6} 4 \, d\theta \).
• Evaluate the integral: Perform the integration, which in this case results in \( 4\theta \Big|_0^{\pi/6} = \frac{2\pi}{3} \).

By following these structured steps, even complex-looking integrals can become straightforward. Attention to detail in each step ensures accuracy and comprehension, helping us apply these concepts across various calculus problems.