Problem 81 - Thomas Calculus

Question

Evaluate the integrals in Exercises \(71-94\) $$ \int_{-1}^{-\sqrt{2} / 2} \frac{d y}{y \sqrt{4 y^{2}-1}} $$

Step-by-Step Solution

Answer

The integral evaluates to \( \frac{\pi}{12} \).

Step 1: Analyze the Integral

The integral given is \( \int_{-1}^{-\sqrt{2} / 2} \frac{d y}{y \sqrt{4 y^{2}-1}} \). Here, the integrand involves a function of the form \( \frac{1}{y \sqrt{4y^2 - 1}} \). This suggests a trigonometric substitution might be useful due to the \( \sqrt{4y^2 - 1} \) term.

Step 2: Determine the Substitution

For integrals involving \( \sqrt{a^2y^2 - b^2} \), a trigonometric substitution can simplify the problem. Here, we consider substituting \( y = \frac{1}{2} \sec(\theta) \), which implies \( dy = \frac{1}{2} \sec(\theta) \tan(\theta) d\theta \). This substitution transforms \( \sqrt{4y^2 - 1} \) into \( \sqrt{4(\frac{1}{4}\sec^2\theta) - 1} = \tan(\theta) \), leveraging the identity \( \sec^2\theta - 1 = \tan^2\theta \).

Step 3: Apply the Substitution

Substitute \( y = \frac{1}{2} \sec(\theta) \) into the integral. The limits change accordingly: when \( y = -1 \), \( \sec(\theta) = -2 \), giving \( \theta = \cos^{-1}(-\frac{1}{2}) = \frac{2\pi}{3} \); when \( y = -\sqrt{2}/2 \), \( \sec(\theta) = -\sqrt{2} \), giving \( \theta = \sec^{-1}(-\sqrt{2}) = \frac{3\pi}{4} \). The integral becomes: \[ \int_{\frac{2\pi}{3}}^{\frac{3\pi}{4}} \frac{\frac{1}{2}\sec(\theta)\tan(\theta)}{\frac{1}{2}\sec(\theta)\tan(\theta)} d\theta. \]

Step 4: Simplify and Integrate

The integrand simplifies to \( 1 \), so the integral becomes \[ \int_{\frac{2\pi}{3}}^{\frac{3\pi}{4}} 1 \, d\theta = \theta \bigg|_{\frac{2\pi}{3}}^{\frac{3\pi}{4}}. \] Thus, evaluate it to get \( \theta \bigg|_{\frac{2\pi}{3}}^{\frac{3\pi}{4}} = \frac{3\pi}{4} - \frac{2\pi}{3}. \)

Step 5: Compute the Final Result

Calculate \( \frac{3\pi}{4} - \frac{2\pi}{3} \) by finding a common denominator. \[ \frac{3\pi}{4} = \frac{9\pi}{12} \quad \text{and} \quad \frac{2\pi}{3} = \frac{8\pi}{12}, \] so the result is \( \frac{9\pi}{12} - \frac{8\pi}{12} = \frac{\pi}{12} \).

Key Concepts

Integral Evaluation

Integral evaluation involves finding the area under a curve represented by a given function. When you see an integral such as \( \int \frac{1}{y \sqrt{4y^2 - 1}} \, dy \), the first step is recognizing the structure of the integrand. This integral is challenging because it contains a complicated square root expression in the denominator: \( \sqrt{4y^2 - 1} \). This suggests using a technique like substitution, which can simplify complex terms.
To handle such integrals effectively, trigonometric substitution is a powerful tool. It turns intricate expressions into manageable forms. In this case, you assess the integral and notice that the expression inside the square root resembles a trigonometric identity, aiding you in deciding a substitution strategy. By identifying the structure, you determine the appropriate substitution formula and then evaluate the integral by simplifying it to a more straightforward form. After substituting and transforming, you calculate the integral using the new variable.
The process of integral evaluation in this context involves transforming variables and recalculating boundaries, attending to the function's limits, and substituting back to express the answer in original terms, ensuring a correct transformative approach has been applied throughout.

Trigonometric Identities

Trigonometric identities are formulas involving trigonometric functions that are always true for every value of the occurring variables. They are essential in simplifying and solving integrals, especially when transformations or substitutions are involved.
In the exercise, we use trigonometric identities to switch from one form to another. By substituting \( y = \frac{1}{2} \sec(\theta) \), we exploit the identity \( \sec^2(\theta) - 1 = \tan^2(\theta) \). This substitution transforms the difficult square root expression \( \sqrt{4y^2 - 1} \) directly into \( \tan(\theta) \), a much simpler form to integrate.

• Understanding these identities allows for transforming more difficult integrals into forms that are straightforward to compute.
• Using \( \sec \) and \( \tan \) is common in these cases due to their relationship with square root expressions in integrals.

Proper handling of these identities simplifies not only the integration process but also the computation of limits and transformations, making it a crucial skill in calculus.

Limit Transformation

Limit transformation is the process of changing the limits of integration when performing a substitution in a definite integral. This step ensures the integral is correctly evaluated over the new variable's values.
In the given problem, when we substitute \( y = \frac{1}{2} \sec(\theta) \), we must adjust the original limits \( y = -1 \) to \( y = -\sqrt{2}/2 \). This requires computing the angles corresponding to these \( y \)-values in terms of \( \theta \).

• For \( y = -1 \), \( \sec(\theta) = -2 \), leading to \( \theta = \cos^{-1}(-\frac{1}{2}) = \frac{2\pi}{3} \).
• For \( y = -\sqrt{2}/2 \), \( \sec(\theta) = -\sqrt{2} \), resulting in \( \theta = \sec^{-1}(-\sqrt{2}) = \frac{3\pi}{4} \).

This changes the integral to operate within the new bounds, ensuring the results remain accurate according to the transformed variable. Through precise calculation of these new limits, the integral is properly evaluated, taking into account the transformed domain of integration.