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...
Related questions
Question
![Given:
\[
\int_0^5 f(x)\, dx = 19 \quad \text{and} \quad \int_5^7 f(x)\, dx = 11
\]
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:**
To evaluate these integrals, we consider the properties of definite integrals.
(a) The integral from 0 to 7 can be split as:
\[
\int_0^7 f(x)\, dx = \int_0^5 f(x)\, dx + \int_5^7 f(x)\, dx = 19 + 11
\]
(b) The integral from 5 to 0 is the negative of the integral from 0 to 5, so:
\[
\int_5^0 f(x)\, dx = -\int_0^5 f(x)\, dx = -19
\]
(c) The integral from 5 to 5 is always zero, because there is no interval:
\[
\int_5^5 f(x)\, dx = 0
\]
(d) For this integral, we can factor out the constant (2) from the integral:
\[
\int_0^5 2f(x)\, dx = 2 \times \int_0^5 f(x)\, dx = 2 \times 19
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe42c0efe-14fe-42af-bc89-aaae6e3d971b%2Fdf6c2c88-c9f3-4bf5-8bc6-963fe557cd87%2Flcrww5_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Given:
\[
\int_0^5 f(x)\, dx = 19 \quad \text{and} \quad \int_5^7 f(x)\, dx = 11
\]
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:**
To evaluate these integrals, we consider the properties of definite integrals.
(a) The integral from 0 to 7 can be split as:
\[
\int_0^7 f(x)\, dx = \int_0^5 f(x)\, dx + \int_5^7 f(x)\, dx = 19 + 11
\]
(b) The integral from 5 to 0 is the negative of the integral from 0 to 5, so:
\[
\int_5^0 f(x)\, dx = -\int_0^5 f(x)\, dx = -19
\]
(c) The integral from 5 to 5 is always zero, because there is no interval:
\[
\int_5^5 f(x)\, dx = 0
\]
(d) For this integral, we can factor out the constant (2) from the integral:
\[
\int_0^5 2f(x)\, dx = 2 \times \int_0^5 f(x)\, dx = 2 \times 19
\]
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