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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please be accurate
![**Problem Statement:**
Evaluate the integral \(\int_{2}^{0} f(x) \, dx\), given that \(\int_{0}^{2} f(x) \, dx = 6\).
**User Input:**
The user inputs the number 12, which is marked incorrect.
**Explanation:**
The problem involves evaluating a definite integral with a reversed interval. The key idea here is understanding how to handle the direction of integration:
- The given condition is \(\int_{0}^{2} f(x) \, dx = 6\).
To find \(\int_{2}^{0} f(x) \, dx\), we use the property of definite integrals where reversing the limits of integration changes the sign:
\[
\int_{2}^{0} f(x) \, dx = -\int_{0}^{2} f(x) \, dx = -6.
\]
Thus, the correct value for \(\int_{2}^{0} f(x) \, dx\) is \(-6\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad35ec9e-c4ff-461f-8442-b935d287c97b%2F30a74ae1-a815-44fe-b1fe-43fcf1fe2578%2Fc76zw4_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Evaluate the integral \(\int_{2}^{0} f(x) \, dx\), given that \(\int_{0}^{2} f(x) \, dx = 6\).
**User Input:**
The user inputs the number 12, which is marked incorrect.
**Explanation:**
The problem involves evaluating a definite integral with a reversed interval. The key idea here is understanding how to handle the direction of integration:
- The given condition is \(\int_{0}^{2} f(x) \, dx = 6\).
To find \(\int_{2}^{0} f(x) \, dx\), we use the property of definite integrals where reversing the limits of integration changes the sign:
\[
\int_{2}^{0} f(x) \, dx = -\int_{0}^{2} f(x) \, dx = -6.
\]
Thus, the correct value for \(\int_{2}^{0} f(x) \, dx\) is \(-6\).
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