Textbook Question
{Use of Tech} Areas of regions Find the area of the region π bounded by the graph of Ζ and the π-axis on the given interval. Graph Ζ and show the region π .
Ζ(π) = 2 β |π| on [ β 2 , 4]
Ch. 5 - Integration
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 5, Problem 5.5.92
Integrals with sinΒ² π and cosΒ² π Evaluate the following integrals.
β« π cosΒ²πΒ² dπ
Verified step by step guidance1
Step 1: Recognize that the integral involves a trigonometric function squared, specifically cosΒ²(πΒ²). To simplify, use the trigonometric identity cosΒ²(u) = (1 + cos(2u)) / 2, where u = πΒ² in this case.
Step 2: Substitute the identity into the integral. The integral becomes β« π * (1 + cos(2πΒ²)) / 2 dπ. Split the integral into two parts: β« π/2 dπ + β« π * cos(2πΒ²)/2 dπ.
Step 3: For the first term, β« π/2 dπ, integrate directly using the power rule for integration: β« π^n dx = (π^(n+1)) / (n+1). This gives (πΒ² / 4).
Step 4: For the second term, β« π * cos(2πΒ²)/2 dπ, use substitution. Let u = 2πΒ², so du = 4π dπ. Rewrite the integral in terms of u: (1/8) β« cos(u) du. The integral of cos(u) is sin(u), so this term becomes (1/8) sin(2πΒ²).
Step 5: Combine the results from both terms. The final expression for the integral is (πΒ² / 4) + (1/8) sin(2πΒ²) + C, where C is the constant of integration.
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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 are true for all values of the variables. Key identities include the Pythagorean identities, such as sinΒ²x + cosΒ²x = 1, and double angle formulas. These identities are essential for simplifying integrals involving sinΒ²x and cosΒ²x, allowing for easier evaluation.
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Integration Techniques
Integration techniques are methods used to find the integral of a function. Common techniques include substitution, integration by parts, and trigonometric substitution. For integrals involving trigonometric functions like cosΒ²x, recognizing patterns and applying appropriate techniques is crucial for finding the antiderivative.
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Definite and Indefinite Integrals
Definite integrals calculate the area under a curve between two specified limits, while indefinite integrals represent a family of functions without limits. Understanding the difference is important when evaluating integrals, as it affects the final result. In this context, knowing whether to apply limits or find a general antiderivative is key to solving the problem.
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Related Practice
Textbook Question
Why can the constant of integration be omitted from the antiderivative when evaluating a definite integral?
Textbook Question
Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.
β«ββΒΉ (πβ1) (πΒ²β2π)β· dπ
Textbook Question
The linear function Ζ(π) = 3 β π is decreasing on the interval [0, 3]. Is its area function for Ζ (with left endpoint 0) increasing or decreasing on the interval [0, 3]? Draw a picture and explain.
Textbook Question
Definite integrals from graphs The figure shows the areas of regions bounded by the graph of Ζ and the π-axis. Evaluate the following integrals.
β«βαΆ |Ζ(π)| dπ
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Textbook Question
Suppose the interval [1, 3] is partitioned into n = 4 subintervals. What is the subinterval length βπ? List the grid points xβ , xβ , xβ , xβ and xβ. Which points are used for the left, right, and midpoint Riemann sums?
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