### Integral Calculations and Area DeterminationConsider the function $ f(x) $, which is defined as follows:- $ f(x) $ is positive when $ 1 < x < 2 $.- $ f(x) $ is negative when $ 2 < x < 3 $.Given the following areas between the graph of $ f(x) $ and the x-axis:- The area from $ x = 1 $ to $ x = 2 $ is 5.- The area from $ x = 2 $ to $ x = 3 $ is 7.We need to determine the various definite integrals of $ f(x) $.1. $\int_{1}^{2} f(x) \, dx = $2. $\int_{2}^{3} f(x) \, dx = $3. $\int_{2}^{1} f(x) \, dx = $4. $\int_{1}^{3} f(x) \, dx = $### Detailed Explanation:**(a) $\int_{1}^{2} f(x) \, dx = $**The function $ f(x) $ is positive between $ x = 1 $ and $ x = 2 $. The given area under the curve in this interval is 5. Thus:\[\int_{1}^{2} f(x) \, dx = 5\]**(b) $\int_{2}^{3} f(x) \, dx = $**In this interval, $ f(x) $ is negative, and the area calculation given is 7. Since the function is negative, we take the negative value of the area:\[\int_{2}^{3} f(x) \, dx = -7\]**(c) $\int_{2}^{1} f(x) \, dx = $**Notice that this is the integral from $ x = 2 $ to $ x = 1 $, which is equal to the negative of the integral from $ x = 1 $ to $ x = 2 $:\[\int_{2}^{1} f(x) \, dx = -\int_{1}^{2} f(x) \, dx = -5\]**(d) \(\

the area from 1 to 2 is 5. so we can write the area from 2 to 3 is 7. so we can write

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Calculus

Function f(x) is positive when 1

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

### Integral Calculations and Area Determination

Consider the function $ f(x) $, which is defined as follows:
- $ f(x) $ is positive when $ 1 < x < 2 $.
- $ f(x) $ is negative when $ 2 < x < 3 $.

Given the following areas between the graph of $ f(x) $ and the x-axis:
- The area from $ x = 1 $ to $ x = 2 $ is 5.
- The area from $ x = 2 $ to $ x = 3 $ is 7.

We need to determine the various definite integrals of $ f(x) $.

1. $\int_{1}^{2} f(x) \, dx = $
2. $\int_{2}^{3} f(x) \, dx = $
3. $\int_{2}^{1} f(x) \, dx = $
4. $\int_{1}^{3} f(x) \, dx = $

### Detailed Explanation:

**(a) $\int_{1}^{2} f(x) \, dx = $**

The function $ f(x) $ is positive between $ x = 1 $ and $ x = 2 $. The given area under the curve in this interval is 5. Thus:

\[
\int_{1}^{2} f(x) \, dx = 5
\]

**(b) $\int_{2}^{3} f(x) \, dx = $**

In this interval, $ f(x) $ is negative, and the area calculation given is 7. Since the function is negative, we take the negative value of the area:

\[
\int_{2}^{3} f(x) \, dx = -7
\]

**(c) $\int_{2}^{1} f(x) \, dx = $**

Notice that this is the integral from $ x = 2 $ to $ x = 1 $, which is equal to the negative of the integral from $ x = 1 $ to $ x = 2 $:

\[
\int_{2}^{1} f(x) \, dx = -\int_{1}^{2} f(x) \, dx = -5
\]

**(d) \(\

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