12. Evaluate (2+ r+3|) dx by interpreting it in terms of areas. Y-3+ 1-4 +3 %3D 2. a dy X dr t 3 dk 11-1+e V =3 4 2. A, +Az A, e 을 .bb .4.3 .12 Az =żbih
12. Evaluate (2+ r+3|) dx by interpreting it in terms of areas. Y-3+ 1-4 +3 %3D 2. a dy X dr t 3 dk 11-1+e V =3 4 2. A, +Az A, e 을 .bb .4.3 .12 Az =żbih
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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![### Calculating Definite Integral Using Area Interpretation
**Problem Statement:**
Evaluate the integral \(\int_{-4}^{2} (2 + |x + 3|) \, dx\) by interpreting it in terms of areas.
---
**Explanation and Solution:**
**Graph Description:**
- The graph is a piecewise function consisting of linear segments, reflecting the absolute value function \(|x + 3|\).
- The function can be broken down into:
- \(y = 2 + |x + 3|\)
**Analysis of the Function:**
1. **Function Analysis:**
- The equation can be expressed as two separate lines:
- For \(x < -3\), \(y = 2 - (x + 3) = -x - 1\)
- For \(x \geq -3\), \(y = 2 + (x + 3) = x + 5\)
2. **Graphical Representation:**
- The graph shows two regions (areas \(A_1\) and \(A_2\)) under the line and above the x-axis within the interval \([-4, 2]\).
- There’s a diagonal line intersecting at \(y = 3\), where the function changes at \(x = -3\).
**Calculation of Areas:**
- **Area \(A_1\) (Triangle from \(x = -4\) to \(x = -3\)):**
- Triangle with base = 4, height = 3
- \(A_1 = \frac{1}{2} \cdot \text{base} \cdot \text{height} = \frac{1}{2} \cdot 4 \cdot 3 = 6\)
- **Area \(A_2\) (Triangle from \(x = -3\) to \(x = 2\)):**
- Triangle with base = 2, height = 2
- \(A_2 = \frac{1}{2} \cdot \text{base} \cdot \text{height} = \frac{1}{2} \cdot 2 \cdot 2 = 2\)
**Total Area:**
- The total area under the curve is the sum of areas \(A_1\) and \(A_](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F17dfd806-082f-4257-81d0-f087a7f9603f%2F623452c0-78b1-4b96-8776-482a8ea0e9b6%2Fx78nne9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Calculating Definite Integral Using Area Interpretation
**Problem Statement:**
Evaluate the integral \(\int_{-4}^{2} (2 + |x + 3|) \, dx\) by interpreting it in terms of areas.
---
**Explanation and Solution:**
**Graph Description:**
- The graph is a piecewise function consisting of linear segments, reflecting the absolute value function \(|x + 3|\).
- The function can be broken down into:
- \(y = 2 + |x + 3|\)
**Analysis of the Function:**
1. **Function Analysis:**
- The equation can be expressed as two separate lines:
- For \(x < -3\), \(y = 2 - (x + 3) = -x - 1\)
- For \(x \geq -3\), \(y = 2 + (x + 3) = x + 5\)
2. **Graphical Representation:**
- The graph shows two regions (areas \(A_1\) and \(A_2\)) under the line and above the x-axis within the interval \([-4, 2]\).
- There’s a diagonal line intersecting at \(y = 3\), where the function changes at \(x = -3\).
**Calculation of Areas:**
- **Area \(A_1\) (Triangle from \(x = -4\) to \(x = -3\)):**
- Triangle with base = 4, height = 3
- \(A_1 = \frac{1}{2} \cdot \text{base} \cdot \text{height} = \frac{1}{2} \cdot 4 \cdot 3 = 6\)
- **Area \(A_2\) (Triangle from \(x = -3\) to \(x = 2\)):**
- Triangle with base = 2, height = 2
- \(A_2 = \frac{1}{2} \cdot \text{base} \cdot \text{height} = \frac{1}{2} \cdot 2 \cdot 2 = 2\)
**Total Area:**
- The total area under the curve is the sum of areas \(A_1\) and \(A_
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