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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Question
![Given:
\[
\int_0^5 f(x) \, dx = 16 \quad \text{and} \quad \int_5^7 f(x) \, dx = 8
\]
Evaluate:
(a) \(\int_0^7 f(x) \, dx\).
(b) \(\int_5^0 f(x) \, dx\).
(c) \(\int_5^5 f(x) \, dx\).
(d) \(\int_0^5 2f(x) \, dx\).
### Explanation:
1. **Problem (a):** Combine the integrals from 0 to 5 and from 5 to 7 to find the integral from 0 to 7.
2. **Problem (b):** Reverse the limits of integration for the integral from 5 to 0, which will change the sign.
3. **Problem (c):** The integral with identical upper and lower limits is always zero.
4. **Problem (d):** Use the property of integrals involving constants to evaluate the integral with the function multiplied by 2.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F862c34ee-e27f-46b4-a2d2-ab5f4ef78243%2F3cd5378c-0fb0-49dd-a076-6e543b153c93%2Fda4k8v_processed.png&w=3840&q=75)
Transcribed Image Text:Given:
\[
\int_0^5 f(x) \, dx = 16 \quad \text{and} \quad \int_5^7 f(x) \, dx = 8
\]
Evaluate:
(a) \(\int_0^7 f(x) \, dx\).
(b) \(\int_5^0 f(x) \, dx\).
(c) \(\int_5^5 f(x) \, dx\).
(d) \(\int_0^5 2f(x) \, dx\).
### Explanation:
1. **Problem (a):** Combine the integrals from 0 to 5 and from 5 to 7 to find the integral from 0 to 7.
2. **Problem (b):** Reverse the limits of integration for the integral from 5 to 0, which will change the sign.
3. **Problem (c):** The integral with identical upper and lower limits is always zero.
4. **Problem (d):** Use the property of integrals involving constants to evaluate the integral with the function multiplied by 2.
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