Textbook Question
Evaluate the integrals in Exercises 1–22.
∫₀^π 8 sin⁴(y) cos²(y) dy
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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 8.4.63
Find the average value of f(x) = (√(x + 1)) / √x on the interval [1, 3].
Verified step by step guidance1
Recall the formula for the average value of a function \(f(x)\) on the interval \([a, b]\):
\[\text{Average value} = \frac{1}{b - a} \int_a^b f(x) \, dx\]
Identify the function and interval: here, \(f(x) = \frac{\sqrt{x + 1}}{\sqrt{x}}\) and the interval is \([1, 3]\). So, \(a = 1\) and \(b = 3\).
Rewrite the function to simplify the integral:
\[f(x) = \frac{\sqrt{x + 1}}{\sqrt{x}} = \sqrt{\frac{x + 1}{x}} = \sqrt{1 + \frac{1}{x}}\]
Set up the integral for the average value:
\[\frac{1}{3 - 1} \int_1^3 \sqrt{1 + \frac{1}{x}} \, dx = \frac{1}{2} \int_1^3 \sqrt{1 + \frac{1}{x}} \, dx\]
To evaluate the integral, consider substitution or algebraic manipulation to simplify the integrand before integrating. After finding the integral, multiply by \(\frac{1}{2}\) to get the average value.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Average Value of a Function
The average value of a function f(x) on an interval [a, b] is given by (1/(b - a)) times the definite integral of f(x) from a to b. It represents the mean height of the function over that interval.
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Average Value of a Function
Definite Integral
A definite integral calculates the net area under the curve of a function between two points a and b. It is essential for finding the total accumulation or average value of a function over an interval.
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Simplifying and Integrating Radical Functions
Functions involving square roots often require algebraic manipulation to simplify before integration. Techniques include rewriting radicals as fractional exponents and applying integration rules for power functions.
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Related Practice
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Textbook Question
Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
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