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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![**Problem Statement:**
Given:
\[
\int_{0}^{2} f(x) \, dx = 3
\]
and
\[
\int_{2}^{5} f(x) \, dx = -9
\]
evaluate the following integrals:
---
**(a)**
\[
\int_{0}^{5} f(x) \, dx
\]
*Solution:*
Calculate the integral from 0 to 5 using the given information.
\[
\int_{0}^{5} f(x) \, dx = \int_{0}^{2} f(x) \, dx + \int_{2}^{5} f(x) \, dx
\]
\[
= 3 + (-9) = -6
\]
---
**(b)**
\[
\int_{5}^{2} f(x) \, dx
\]
*Solution:*
Reverse the integral. Reversing the limits of integration changes the sign.
\[
\int_{5}^{2} f(x) \, dx = -\int_{2}^{5} f(x) \, dx
\]
\[
= -(-9) = 9
\]
---
**(c)**
\[
\int_{3}^{3} f(x) \, dx
\]
*Solution:*
The integral of any function over an interval of zero length is zero.
\[
\int_{3}^{3} f(x) \, dx = 0
\]
---
**(d)**
\[
\int_{2}^{5} -8f(x) \, dx
\]
*Solution:*
Factor out the constant (-8) from the integral.
\[
\int_{2}^{5} -8f(x) \, dx = -8 \int_{2}^{5} f(x) \, dx
\]
\[
= -8(-9) = 72
\]
---
These solutions use the properties of definite integrals, including linearity and changes in limits of integration.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F25721e20-facb-4d0d-8f2d-fb325bdcbfad%2F05682ac1-9fae-4f5b-888f-a625e9a90786%2F5u3t5x_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Given:
\[
\int_{0}^{2} f(x) \, dx = 3
\]
and
\[
\int_{2}^{5} f(x) \, dx = -9
\]
evaluate the following integrals:
---
**(a)**
\[
\int_{0}^{5} f(x) \, dx
\]
*Solution:*
Calculate the integral from 0 to 5 using the given information.
\[
\int_{0}^{5} f(x) \, dx = \int_{0}^{2} f(x) \, dx + \int_{2}^{5} f(x) \, dx
\]
\[
= 3 + (-9) = -6
\]
---
**(b)**
\[
\int_{5}^{2} f(x) \, dx
\]
*Solution:*
Reverse the integral. Reversing the limits of integration changes the sign.
\[
\int_{5}^{2} f(x) \, dx = -\int_{2}^{5} f(x) \, dx
\]
\[
= -(-9) = 9
\]
---
**(c)**
\[
\int_{3}^{3} f(x) \, dx
\]
*Solution:*
The integral of any function over an interval of zero length is zero.
\[
\int_{3}^{3} f(x) \, dx = 0
\]
---
**(d)**
\[
\int_{2}^{5} -8f(x) \, dx
\]
*Solution:*
Factor out the constant (-8) from the integral.
\[
\int_{2}^{5} -8f(x) \, dx = -8 \int_{2}^{5} f(x) \, dx
\]
\[
= -8(-9) = 72
\]
---
These solutions use the properties of definite integrals, including linearity and changes in limits of integration.
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