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
![**Step 1**
We are concerned with the segment of the line \( y = \frac{5}{2}x - 10 \) that begins at \( (0, -10) \) and ends at \( \left( 7, \frac{15}{2} \right) \).
**Graph Explanation:**
The graph displays a coordinate plane with a line segment starting at the point \( (0, -10) \) and ending at \( \left( 7, \frac{15}{2} \right) \). The line crosses the x-axis to form a triangular region with the axes. The triangular area above the x-axis is shaded in gray.
**Step 2**
The integral \( \int_{0}^{7} \left( \frac{5}{2}x - 10 \right) \, dx \) can be interpreted as the area of the triangle above the x-axis minus the area of the triangle below the x-axis.
The formula for the area of the triangle below the x-axis is \( \frac{1}{2} bh = \) [blank field] with an incorrect mark, indicating an error or missing calculation. The area of the upper triangle also has a [blank field] with an incorrect mark, indicating the same.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F25226f33-2fc3-40c9-94fe-a259b74d1925%2Fecf6a6f2-08e3-4aa2-8a5a-3e2a06264618%2Fxyj6k8p_processed.png&w=3840&q=75)
Transcribed Image Text:**Step 1**
We are concerned with the segment of the line \( y = \frac{5}{2}x - 10 \) that begins at \( (0, -10) \) and ends at \( \left( 7, \frac{15}{2} \right) \).
**Graph Explanation:**
The graph displays a coordinate plane with a line segment starting at the point \( (0, -10) \) and ending at \( \left( 7, \frac{15}{2} \right) \). The line crosses the x-axis to form a triangular region with the axes. The triangular area above the x-axis is shaded in gray.
**Step 2**
The integral \( \int_{0}^{7} \left( \frac{5}{2}x - 10 \right) \, dx \) can be interpreted as the area of the triangle above the x-axis minus the area of the triangle below the x-axis.
The formula for the area of the triangle below the x-axis is \( \frac{1}{2} bh = \) [blank field] with an incorrect mark, indicating an error or missing calculation. The area of the upper triangle also has a [blank field] with an incorrect mark, indicating the same.
![**Title**: Evaluating Integrals Through Area Interpretation
**Objective**: Learn to evaluate the integral by interpreting it in terms of geometric areas.
**Problem Statement**:
Evaluate the integral by interpreting it in terms of areas.
\[
\int_{0}^{7}\left(\frac{5}{2}x - 10\right) \, dx
\]
**Explanation**:
This integral represents the area under the curve described by the equation \(\frac{5}{2}x - 10\), from \(x = 0\) to \(x = 7\). The expression \(\frac{5}{2}x - 10\) is a linear function, which implies the graph of the function is a straight line.
To solve this, you will:
1. Identify major points by solving \(\frac{5}{2}x - 10 = 0\) to find where the line crosses the x-axis.
2. Recognize the geometric shapes formed under the line within the interval \(0\) to \(7\).
3. Calculate the areas of these shapes, interpret them in terms of basic geometric areas (e.g., triangles, rectangles).
The area found will be the net area— account for areas above and below the x-axis accordingly. In this case, a positive and/or negative value will indicate whether the respective sections of areas fall above or below the axis.
**Conclusion**:
Understanding integrals through the lens of geometric areas supplies a visual and conceptual method of computation, enhancing comprehension for complex integral problems.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F25226f33-2fc3-40c9-94fe-a259b74d1925%2Fecf6a6f2-08e3-4aa2-8a5a-3e2a06264618%2Fic7z56p_processed.png&w=3840&q=75)
Transcribed Image Text:**Title**: Evaluating Integrals Through Area Interpretation
**Objective**: Learn to evaluate the integral by interpreting it in terms of geometric areas.
**Problem Statement**:
Evaluate the integral by interpreting it in terms of areas.
\[
\int_{0}^{7}\left(\frac{5}{2}x - 10\right) \, dx
\]
**Explanation**:
This integral represents the area under the curve described by the equation \(\frac{5}{2}x - 10\), from \(x = 0\) to \(x = 7\). The expression \(\frac{5}{2}x - 10\) is a linear function, which implies the graph of the function is a straight line.
To solve this, you will:
1. Identify major points by solving \(\frac{5}{2}x - 10 = 0\) to find where the line crosses the x-axis.
2. Recognize the geometric shapes formed under the line within the interval \(0\) to \(7\).
3. Calculate the areas of these shapes, interpret them in terms of basic geometric areas (e.g., triangles, rectangles).
The area found will be the net area— account for areas above and below the x-axis accordingly. In this case, a positive and/or negative value will indicate whether the respective sections of areas fall above or below the axis.
**Conclusion**:
Understanding integrals through the lens of geometric areas supplies a visual and conceptual method of computation, enhancing comprehension for complex integral problems.
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