Textbook Question
Area: Find the area between the x-axis and the curve y = √(1 + cos 4x), for 0 ≤ x ≤ π.
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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 8, Problem 8.3.18
Evaluate the integrals in Exercises 1–22.
∫₀^π 8cos⁴(2πx) dx
Verified step by step guidance1
Recognize that the integral is \( \int_0^{\pi} 8 \cos^4(2\pi x) \, dx \). The constant 8 can be factored out of the integral, so rewrite it as \( 8 \int_0^{\pi} \cos^4(2\pi x) \, dx \).
Use the power-reduction formula to express \( \cos^4(\theta) \) in terms of cosines of multiple angles. Recall that \( \cos^4(\theta) = \left( \cos^2(\theta) \right)^2 \) and \( \cos^2(\theta) = \frac{1 + \cos(2\theta)}{2} \).
Substitute \( \cos^2(2\pi x) = \frac{1 + \cos(4\pi x)}{2} \) into the expression and then square it to get \( \cos^4(2\pi x) = \left( \frac{1 + \cos(4\pi x)}{2} \right)^2 \).
Expand the squared term to get \( \cos^4(2\pi x) = \frac{1}{4} \left( 1 + 2\cos(4\pi x) + \cos^2(4\pi x) \right) \). Then apply the power-reduction formula again to \( \cos^2(4\pi x) \) to express it in terms of cosines of multiple angles.
Rewrite the integral as a sum of integrals of cosines with different frequencies, integrate each term over \( 0 \) to \( \pi \), and then multiply the result by 8. Remember to use the fact that \( \int_0^{\pi} \cos(kx) \, dx = 0 \) for nonzero integer multiples \( k \) of \( \pi \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definite Integrals
A definite integral calculates the net area under a curve between two specific limits. It is represented as ∫_a^b f(x) dx, where a and b are the lower and upper bounds. Evaluating definite integrals often involves finding an antiderivative and then applying the Fundamental Theorem of Calculus.
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Definition of the Definite Integral
Trigonometric Power Reduction
Power reduction formulas transform powers of trigonometric functions into expressions involving multiple angles with lower powers. For example, cos⁴(θ) can be rewritten using cos(2θ) and cos(4θ) terms, simplifying integration by converting powers into sums of cosines.
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Integration of Cosine Functions
Integrating cosine functions involves using standard integral formulas, such as ∫cos(kx) dx = (1/k) sin(kx) + C. When combined with definite limits, evaluating these integrals requires substituting the limits into the antiderivative and calculating the difference.
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Related Practice
Textbook Question
Evaluate the integrals in Exercises 39–54.
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The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
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Evaluate the integrals in Exercises 33–52.
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Textbook Question
Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
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