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Magazine Chapter 7: Problem 34
Evaluate each integral. \(\int \frac{\sqrt{9-y^{2}}}{y} d y\)
Short Answer
Expert verified
The integral is \(3 \ln|y| - y + C\).
Step by step solution
01
Substitute Trigonometric Function
To solve the integral \( \int \frac{\sqrt{9-y^{2}}}{y} dy \), let's use a trigonometric substitution. Set \( y = 3 \sin \theta \). Then, \( dy = 3 \cos \theta \, d\theta \). The integrand becomes \( \frac{\sqrt{9 - (3\sin\theta)^2}}{3 \sin \theta} \cdot 3\cos\theta \, d\theta \).
02
Simplify the Integrand
Substitute \( y = 3 \sin \theta \) into the integrand: \[ \int \frac{3\cos\theta \sqrt{9 - 9\sin^2\theta}}{3\sin\theta} \, d\theta = \int \frac{3\cos\theta \cdot 3\cos\theta}{3\sin\theta} \, d\theta = \int \frac{9\cos^2\theta}{3\sin\theta} \, d\theta. \] Simplifying, this becomes \[ 3 \int \cot\theta\cos\theta \, d\theta \].
03
Apply Trigonometric Identity
Recall that \( \cot\theta = \frac{\cos\theta}{\sin\theta} \), so \( \cot\theta\cos\theta = \frac{\cos^2\theta}{\sin\theta} \). The integral is \[ 3 \int \frac{\cos^2\theta}{\sin\theta} \, d\theta = 3 \int \cot\theta \, d\theta - 3 \int \cos\theta \, d\theta. \]
04
Integrate Each Term
Integrate each term separately. The integral of \( \cot\theta \) is \( \ln|\sin\theta| \), and the integral of \( \cos\theta \) is \( \sin\theta \). Thus, \[ 3 \left( \ln|\sin\theta| - \sin\theta \right) + C, \] where \( C \) is the constant of integration.
05
Convert Back to y
Substitute back \( y = 3 \sin\theta \) to convert the solution in terms of \( y \). Hence, \( \sin\theta = \frac{y}{3} \), and \( \cos^2\theta = 1 - \sin^2\theta = \frac{9-y^2}{9} \), so \( \cos\theta = \frac{\sqrt{9-y^2}}{3} \). The solution becomes \[ 3 \left( \ln\left|\frac{y}{3}\right| - \frac{y}{3} \right) + C \].
06
Simplify the Expression
Simplify the final expression by combining the constant coefficients and terms. Thus, the solution is \[ 3 \ln|y| - 3\ln3 - y + C. \] The constant \(-3\ln3\) can be absorbed into the constant of integration, resulting in the final expression: \[ 3 \ln|y| - y + C. \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Integration Techniques
Integrating using trigonometric substitution is a smart technique for solving integrals with expressions like \( \sqrt{a^2 - x^2} \). Choosing the right substitution leverages trigonometric identities to simplify the problem.
- First, you identify which part of the integral suggests a trigonometric substitution. Common expressions like \( 9 - y^2 \) hint at using \( y = 3\sin\theta \).
- Once substituted, the problem shifts into a trigonometric framework, often transforming complex algebraic expressions into more straightforward trigonometric ones.
- Remember to change the differential accordingly鈥攚hen substituting \( y = 3 \sin \theta \), then \( dy = 3 \cos \theta \, d\theta \).
Definite and Indefinite Integrals
Understanding the difference between definite and indefinite integrals is crucial. Let's break it down.
- An indefinite integral, as in this exercise, represents a family of functions and includes a constant \( C \), because integration is the reverse process of differentiation.
- Definite integrals, on the other hand, compute the area under the curve between two limits, giving a finite number, thereby eliminating the constant \( C \).
- In this exercise, the expression \( 3 \ln|y| - y + C \) represents an indefinite integral solved using substitution.
Trigonometric Identities
Trigonometric identities were a key player in simplifying the given integral.
- The identity \( \sin^2 \theta + \cos^2 \theta = 1 \) was useful in transforming \( \cos \theta \) expressions.
- With the substitution \( y = 3 \sin \theta \), notice how identities help reduce the complexity: \( \cos^2 \theta = 1 - \sin^2 \theta \).
- Coth and its manipulation is evident in rewriting \( \frac{\cos^2 \theta}{\sin \theta} \), leading to easier integrations.
