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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![### Calculus Problem: Definite Integrals
Given that
\[ \int_{1}^{3} f(x) \, dx = \frac{35}{11} \]
What is
\[ \int_{3}^{1} f(w) \, dw ? \]
To find the answer, enter it in the box below:
**Answer =** [_______]
#### Explanation:
1. The problem presents a definite integral from 1 to 3 of a function \( f(x) \) and provides its value as \( \frac{35}{11} \).
2. It then asks for the value of the definite integral of \( f(w) \) from 3 to 1.
3. Note that reversing the limits of integration changes the sign of the integral.
\[
\int_{a}^{b} f(x) \, dx = - \int_{b}^{a} f(x) \, dx
\]
Therefore,
\[
\int_{3}^{1} f(w) \, dw = - \int_{1}^{3} f(x) \, dx
\]
Given that \( \int_{1}^{3} f(x) \, dx = \frac{35}{11} \),
\[
\int_{3}^{1} f(w) \, dw = -\frac{35}{11}
\]
### Answer
-3.1818](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc3c07474-0773-48a9-bf93-2a357cb5d209%2F391e6bc4-4cad-4fe7-ad27-525ff0a8d18e%2Fw46jt2o_processed.png&w=3840&q=75)
Transcribed Image Text:### Calculus Problem: Definite Integrals
Given that
\[ \int_{1}^{3} f(x) \, dx = \frac{35}{11} \]
What is
\[ \int_{3}^{1} f(w) \, dw ? \]
To find the answer, enter it in the box below:
**Answer =** [_______]
#### Explanation:
1. The problem presents a definite integral from 1 to 3 of a function \( f(x) \) and provides its value as \( \frac{35}{11} \).
2. It then asks for the value of the definite integral of \( f(w) \) from 3 to 1.
3. Note that reversing the limits of integration changes the sign of the integral.
\[
\int_{a}^{b} f(x) \, dx = - \int_{b}^{a} f(x) \, dx
\]
Therefore,
\[
\int_{3}^{1} f(w) \, dw = - \int_{1}^{3} f(x) \, dx
\]
Given that \( \int_{1}^{3} f(x) \, dx = \frac{35}{11} \),
\[
\int_{3}^{1} f(w) \, dw = -\frac{35}{11}
\]
### Answer
-3.1818
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