Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Title: Finding the Average Value of a Trigonometric Function**
**Objective:**
Learn how to find the average value of the function over a specified interval.
**Problem Statement:**
Find the average value of the following function on the given interval:
\[ f(x) = \frac{\pi}{4} \sin(x) \quad \text{on} \quad [0, \pi] \]
**Explanation:**
To find the average value of a continuous function \( f(x) \) over an interval \([a, b]\), use the formula:
\[
\text{Average value of } f(x) = \frac{1}{b-a} \int_{a}^{b} f(x) \, dx
\]
**Application:**
In this problem, apply the formula to \( f(x) = \frac{\pi}{4} \sin(x) \).
1. The interval is \([0, \pi]\).
2. Substitute \( a = 0 \) and \( b = \pi \) into the formula.
\[
\text{Average value} = \frac{1}{\pi - 0} \int_{0}^{\pi} \frac{\pi}{4} \sin(x) \, dx
\]
3. Simplify and compute the integral to find the average value.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4f4fb34f-2a47-4fc8-b888-53dcc6627b0f%2F275c31de-6cf0-4c6c-8e5d-3c4d31797431%2Ffhnjeaf.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Finding the Average Value of a Trigonometric Function**
**Objective:**
Learn how to find the average value of the function over a specified interval.
**Problem Statement:**
Find the average value of the following function on the given interval:
\[ f(x) = \frac{\pi}{4} \sin(x) \quad \text{on} \quad [0, \pi] \]
**Explanation:**
To find the average value of a continuous function \( f(x) \) over an interval \([a, b]\), use the formula:
\[
\text{Average value of } f(x) = \frac{1}{b-a} \int_{a}^{b} f(x) \, dx
\]
**Application:**
In this problem, apply the formula to \( f(x) = \frac{\pi}{4} \sin(x) \).
1. The interval is \([0, \pi]\).
2. Substitute \( a = 0 \) and \( b = \pi \) into the formula.
\[
\text{Average value} = \frac{1}{\pi - 0} \int_{0}^{\pi} \frac{\pi}{4} \sin(x) \, dx
\]
3. Simplify and compute the integral to find the average value.
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