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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![### Understanding Integral Bounds in Calculus
When given a function \( f(x) \) that is bounded between two values over a specified range, we can also determine the bounds for its definite integral over that range.
For instance, consider the inequality:
\[ 10 \leq f(x) \leq 20 \]
We want to find the bounds for the integral of \( f(x) \) from \( x = 3 \) to \( x = 6 \). In mathematical terms, this can be written as:
\[ \int_3^6 f(x) \, dx \]
To find those bounds, we need to consider the minimum and maximum values \( f(x) \) can take over the interval \([3, 6]\).
- The minimum value \( f(x) \) can take is 10.
- The maximum value \( f(x) \) can take is 20.
Using these bounds, we can establish the following inequalities for the integral:
\[ \int_3^6 10 \, dx \leq \int_3^6 f(x) \, dx \leq \int_3^6 20 \, dx \]
Calculating these definite integrals:
\[ \int_3^6 10 \, dx = 10 \times (6 - 3) = 10 \times 3 = 30 \]
\[ \int_3^6 20 \, dx = 20 \times (6 - 3) = 20 \times 3 = 60 \]
Thus, the integral \( \int_3^6 f(x) \, dx \) is bounded by:
\[ 30 \leq \int_3^6 f(x) \, dx \leq 60 \]
This relationship helps us understand that even if \( f(x) \) varies within the given bounds, the value of its integral over the specified range will also be bounded within a predictable range.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff01f17a1-5f0e-4e74-91b1-8cce2d6624f8%2F19c0e2b4-a866-4550-a55f-5b74b2566ebf%2F21ew8pm.png&w=3840&q=75)
Transcribed Image Text:### Understanding Integral Bounds in Calculus
When given a function \( f(x) \) that is bounded between two values over a specified range, we can also determine the bounds for its definite integral over that range.
For instance, consider the inequality:
\[ 10 \leq f(x) \leq 20 \]
We want to find the bounds for the integral of \( f(x) \) from \( x = 3 \) to \( x = 6 \). In mathematical terms, this can be written as:
\[ \int_3^6 f(x) \, dx \]
To find those bounds, we need to consider the minimum and maximum values \( f(x) \) can take over the interval \([3, 6]\).
- The minimum value \( f(x) \) can take is 10.
- The maximum value \( f(x) \) can take is 20.
Using these bounds, we can establish the following inequalities for the integral:
\[ \int_3^6 10 \, dx \leq \int_3^6 f(x) \, dx \leq \int_3^6 20 \, dx \]
Calculating these definite integrals:
\[ \int_3^6 10 \, dx = 10 \times (6 - 3) = 10 \times 3 = 30 \]
\[ \int_3^6 20 \, dx = 20 \times (6 - 3) = 20 \times 3 = 60 \]
Thus, the integral \( \int_3^6 f(x) \, dx \) is bounded by:
\[ 30 \leq \int_3^6 f(x) \, dx \leq 60 \]
This relationship helps us understand that even if \( f(x) \) varies within the given bounds, the value of its integral over the specified range will also be bounded within a predictable range.
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