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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![### Integral of f(x) within the Interval [2, 5]
**Question:**
What is the integral of f(x) within the interval [2, 5]?
**Graph Analysis:**
The graph of the function f(x) is represented in green. It shows the values of f(x) on a coordinate plane with the X-axis and Y-axis labeled accordingly.
- On the X-axis, the key points of interest for the interval [2, 5] are marked.
- On the Y-axis, the function values appear significant at points 2, 4, and 6.
The function f(x) exhibits two distinct segments:
1. A horizontal line segment from (2, 2) to (4, 2).
2. An inclined line segment from (4, 2) to (5, 6).
To find the integral, we need to calculate the area under the curve within the given interval.
**Calculation:**
1. From x = 2 to x = 4, the area is a rectangle:
\[
\text{Area}_1 = \text{width} \times \text{height} = (4 - 2) \times 2 = 4
\]
2. From x = 4 to x = 5, the area is a trapezoid:
\[
\text{Area}_2 = \frac{1}{2} \times (\text{Base}_1 + \text{Base}_2) \times \text{height} = \frac{1}{2} \times (2 + 6) \times (5 - 4) = \frac{1}{2} \times 8 \times 1 = 4
\]
Adding these areas together:
\[
\text{Total Area} = \text{Area}_1 + \text{Area}_2 = 4 + 4 = 8
\]
However, our preset multiple-choice answers seem inaccurate based on our calculation, indicating a possible error or misjudgment in provided choices.
**Choices:**
- ⃝ Area = 11.0
- ⃝ Area = 9.5
- ⃝ Area = 8.5
- ⃝ Area = 7.5
**Conclusion:**
Given the discrepancy, it is implied potential errors exist in the graph evaluation or the question](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Feb401025-c6b7-46e8-825c-86d3393fab5e%2Fa11d4830-e5ab-47dc-8b8f-674d6d308c4c%2Flwks7g_processed.png&w=3840&q=75)
Transcribed Image Text:### Integral of f(x) within the Interval [2, 5]
**Question:**
What is the integral of f(x) within the interval [2, 5]?
**Graph Analysis:**
The graph of the function f(x) is represented in green. It shows the values of f(x) on a coordinate plane with the X-axis and Y-axis labeled accordingly.
- On the X-axis, the key points of interest for the interval [2, 5] are marked.
- On the Y-axis, the function values appear significant at points 2, 4, and 6.
The function f(x) exhibits two distinct segments:
1. A horizontal line segment from (2, 2) to (4, 2).
2. An inclined line segment from (4, 2) to (5, 6).
To find the integral, we need to calculate the area under the curve within the given interval.
**Calculation:**
1. From x = 2 to x = 4, the area is a rectangle:
\[
\text{Area}_1 = \text{width} \times \text{height} = (4 - 2) \times 2 = 4
\]
2. From x = 4 to x = 5, the area is a trapezoid:
\[
\text{Area}_2 = \frac{1}{2} \times (\text{Base}_1 + \text{Base}_2) \times \text{height} = \frac{1}{2} \times (2 + 6) \times (5 - 4) = \frac{1}{2} \times 8 \times 1 = 4
\]
Adding these areas together:
\[
\text{Total Area} = \text{Area}_1 + \text{Area}_2 = 4 + 4 = 8
\]
However, our preset multiple-choice answers seem inaccurate based on our calculation, indicating a possible error or misjudgment in provided choices.
**Choices:**
- ⃝ Area = 11.0
- ⃝ Area = 9.5
- ⃝ Area = 8.5
- ⃝ Area = 7.5
**Conclusion:**
Given the discrepancy, it is implied potential errors exist in the graph evaluation or the question
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