Textbook Question
Area: Find the area enclosed by the ellipse x²/a² + y²/b² = 1.
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Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 8, Problem 8.3.60
Exercises 59–64 require the use of various trigonometric identities before you evaluate the integrals.
∫ cos²(2θ) sin(θ) dθ
Verified step by step guidance1
Recognize that the integral involves \( \cos^2(2\theta) \) and \( \sin(\theta) \), and that simplifying \( \cos^2(2\theta) \) using a trigonometric identity will make the integral easier to evaluate.
Use the power-reduction identity for cosine squared: \( \cos^2(x) = \frac{1 + \cos(2x)}{2} \). Applying this to \( \cos^2(2\theta) \) gives \( \cos^2(2\theta) = \frac{1 + \cos(4\theta)}{2} \).
Rewrite the integral by substituting the identity: \[ \int \cos^2(2\theta) \sin(\theta) \, d\theta = \int \frac{1 + \cos(4\theta)}{2} \sin(\theta) \, d\theta. \]
Distribute \( \sin(\theta) \) inside the integral: \[ \int \frac{1}{2} \sin(\theta) \, d\theta + \int \frac{\cos(4\theta)}{2} \sin(\theta) \, d\theta. \]
Evaluate each integral separately. The first integral is straightforward, while the second integral \( \int \cos(4\theta) \sin(\theta) \, d\theta \) may require using product-to-sum identities or substitution to simplify before integrating.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. They simplify expressions and integrals by transforming products or powers of sine and cosine into sums or simpler forms, such as using the power-reduction or double-angle formulas.
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Verifying Trig Equations as Identities
Integration of Trigonometric Functions
Integrating trigonometric functions involves applying standard integral formulas and substitution techniques. Recognizing when to use identities to rewrite the integrand into a more manageable form is essential for evaluating integrals involving powers or products of sine and cosine.
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6:04Introduction to Trigonometric Functions
Substitution Method in Integration
The substitution method simplifies integrals by changing variables to reduce complexity. When the integrand contains composite functions like cos²(2θ), substituting an inner function or using identities can transform the integral into a basic form that is easier to evaluate.
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07:33Euler's Method
Related Practice
Textbook Question
Evaluate the integrals in Exercises 1–14.
∫ 5 dx / √(25x² - 9), where x > 3/5
Textbook Question
Evaluate the integrals in Exercises 39–54.
∫ sin(θ) dθ / (cos²θ + cos θ - 2)
Textbook Question
In Exercises 17–20, express the integrand as a sum of partial fractions and evaluate the integrals.
∫ (x² dx) / ((x - 1)(x² + 2x + 1))
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Textbook Question
In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.
∫ from 0 to 1 of (dt / (t - sin t))
(Hint: t ≥ sin t for t ≥ 0)
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Textbook Question
In Exercises 9–16, express the integrand as a sum of partial fractions and evaluate the integrals.
∫ (2x + 1) / (x² - 7x + 12) dx
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