Problem 27 - Thomas Calculus

Question

Evaluate each integral in Exercises \(1-36\) by using a substitution to reduce it to standard form. $$ \int_{0}^{1 / 6} \frac{d x}{\sqrt{1-9 x^{2}}} $$

Step-by-Step Solution

Answer

The integral evaluates to \( \frac{\pi}{18} \).

Step 1: Identify a suitable substitution

To solve the integral \( \int_{0}^{1/6} \frac{dx}{\sqrt{1-9x^2}} \), we need to use a substitution that simplifies the expression under the square root. Notice that the form \( 1 - 9x^2 \) resembles the Pythagorean identity \( \sin^2\theta + \cos^2\theta = 1 \). Thus, we choose \( x = \frac{1}{3} \sin \theta \). Then, \( dx = \frac{1}{3} \cos \theta \, d\theta \).

Step 2: Change limits of integration

Convert the integral limits from \( x \) to \( \theta \). When \( x = 0 \), \( \frac{1}{3} \sin \theta = 0 \) giving \( \theta = 0 \). When \( x = \frac{1}{6} \), \( \frac{1}{3} \sin \theta = \frac{1}{6} \) gives \( \sin \theta = \frac{1}{2} \). At this point, \( \theta = \frac{\pi}{6} \).

Step 3: Substitute and simplify the integral

Substituting \( x = \frac{1}{3} \sin \theta \) into the integral transforms it into \( \int_{0}^{\pi/6} \frac{\frac{1}{3} \cos \theta \, d\theta}{\sqrt{1 - 9\left(\frac{1}{3} \sin \theta\right)^2}} \). Simplifying the square root gives \( \sqrt{1 - \sin^2 \theta} = \cos \theta \). The integral then reduces to \( \int_{0}^{\pi/6} \frac{\frac{1}{3} \cos \theta \, d\theta}{\cos \theta} \).

Step 4: Evaluate the integral

The expression simplifies to \( \int_{0}^{\pi/6} \frac{1}{3} \, d\theta \). Integrating with respect to \( \theta \), we get \( \frac{1}{3} [\theta]_{0}^{\pi/6} = \frac{1}{3} \left( \frac{\pi}{6} - 0 \right) \).

Step 5: Compute the final answer

Calculating the result from the integral, we find \( \frac{1}{3} \times \frac{\pi}{6} = \frac{\pi}{18} \). Hence, the value of the integral \( \int_{0}^{1/6} \frac{dx}{\sqrt{1-9x^2}} \) is \( \frac{\pi}{18} \).

Key Concepts

Substitution Method

The substitution method is a powerful technique used in calculus to simplify and solve integrals. It involves replacing a complicated expression with a simpler one using a substitution variable. This can make the integration process more manageable, especially when dealing with complex or unfamiliar functions.

Here's how it works:

• Identify a suitable substitution that can simplify the integral. This often involves using trigonometric identities or algebraic manipulations.
• Find the derivative of the substitution variable and express the original differential in terms of the new variable.
• Change the limits of integration if the integral is definite, adapting them to align with the substitution.
• Perform the integration over the new variable, which should be simpler.
• Revert to the original variable if necessary, using the inverse of the substitution.

In our example, using the substitution method involved switching from an integral with respect to \( x \) to an integral with respect to \( \theta \), allowing for straightforward integration.

Trigonometric Substitution

Trigonometric substitution is a special case of substitution used in integration when the integrand contains expressions like \( a^2 - x^2 \), \( a^2 + x^2 \), or \( x^2 - a^2 \). These expressions often resemble the Pythagorean identity, leading to the employment of trigonometric identities.

The method relies on using trigonometric functions like sine, cosine, or tangent to transform the integral into a more straightforward form. Here is the process:

• Identify the form of the expression; for example, \( 1 - x^2 \) is similar to \( \sin^2\theta + \cos^2\theta = 1 \).
• Choose an appropriate trigonometric identity, such as \( x = a \sin\theta \), to simplify the expression.
• Replace the differential and change the limits if the integral is definite.
• Simplify and integrate using the simple trigonometric expression.
• Convert back to the original variable after integrating.

In the exercise, the substitution \( x = \frac{1}{3} \sin\theta \) allowed the reduction of the complex integral into a simpler one, using the identity that \( \sqrt{1 - \sin^2\theta} \) becomes \( \cos\theta \).

Definite Integral

The definite integral is a fundamental concept in calculus that represents the accumulation of quantities, such as area under a curve, between two limits. It is denoted as \( \int_{a}^{b} f(x) \, dx \), where \( a \) and \( b \) are the integration limits.

The evaluation of a definite integral using substitution involves several key steps:

• Choose a substitution to transform the integral into an easier form.
• Modify the integration limits to correspond with the new variable.
• Integrate the simplified expression.
• Substitute back and evaluate using the original limits, or use adjusted limits with the substitution.

In the solved problem, once the substitution \( x = \frac{1}{3} \sin\theta \) was implemented, the new limits became \( 0 \) and \( \frac{\pi}{6} \), reflecting the transformed variable. This substitution facilitated computing the definite integral effortlessly.

Pythagorean Identity

The Pythagorean identity is an essential trigonometric identity used in various calculus applications, particularly integration. It states that for any angle \( \theta \), \( \sin^2\theta + \cos^2\theta = 1 \). This identity can be a valuable tool in simplifying expressions inside integrals that contain square roots.

In context of trigonometric substitution, using the Pythagorean identity helps reduce complex expressions. Here’s how it applied to the exercise:

• The expression \( 1 - 9x^2 \) was analogous to \( \cos^2\theta \) by choosing \( x = \frac{1}{3} \sin\theta \).
• This transformation allowed the complex square root \( \sqrt{1 - 9x^2} \) to simplify to \( \cos\theta \), using the identity.
• By substituting, the integral turned into a form that was straightforward to evaluate.

Understanding and applying this identity can make such calculus problems much easier to tackle, emphasizing its significant role in integration strategies.