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 the Integral of a Function from its Graph
**Problem Statement:**
Given the graph of \( f(x) \) below, find \( \int_{-1}^{8} f(x) \, dx \).
**Instructions:**
Find the exact value and provide no decimal entries.
\[ \int_{-1}^{8} f(x) \, dx = \underline{\quad\quad\quad} \]
---
**Graph Description:**
The graph is presented on a coordinate grid with both x-axis and y-axis ranging from -3 to 9. The function \( f(x) \) depicted by the curve consists of two parts:
1. **From \( x = -1 \) to \( x = 5 \):**
- The curve is a downward-facing parabola that reaches its minimum at approximately \( (2, -3) \).
- It starts above the x-axis, dips below, and returns to meet the x-axis at \( x = 5 \).
2. **From \( x = 5 \) to \( x = 8 \):**
- The curve becomes linear, rising steeply from the point at \( (5, 0) \) to \( (8, 6) \).
**Integration Approach:**
To find the area under the curve, calculate the integral of each segment:
- **Parabolic Segment (from \( x = -1 \) to \( x = 5 \)):** Calculate the area below the x-axis and subtract it from the area above the axis.
- **Linear Segment (from \( x = 5 \) to \( x = 8 \)):** Calculate the area of the triangle formed by the line and the x-axis.
Calculate these areas accurately using geometric shapes and add them to find \( \int_{-1}^{8} f(x) \, dx \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fec70a3af-7012-41b5-84d0-077491201fef%2F79a34ce0-d14f-47da-be78-da2e57a9e71f%2Fspsr1j_processed.png&w=3840&q=75)
Transcribed Image Text:### Calculating the Integral of a Function from its Graph
**Problem Statement:**
Given the graph of \( f(x) \) below, find \( \int_{-1}^{8} f(x) \, dx \).
**Instructions:**
Find the exact value and provide no decimal entries.
\[ \int_{-1}^{8} f(x) \, dx = \underline{\quad\quad\quad} \]
---
**Graph Description:**
The graph is presented on a coordinate grid with both x-axis and y-axis ranging from -3 to 9. The function \( f(x) \) depicted by the curve consists of two parts:
1. **From \( x = -1 \) to \( x = 5 \):**
- The curve is a downward-facing parabola that reaches its minimum at approximately \( (2, -3) \).
- It starts above the x-axis, dips below, and returns to meet the x-axis at \( x = 5 \).
2. **From \( x = 5 \) to \( x = 8 \):**
- The curve becomes linear, rising steeply from the point at \( (5, 0) \) to \( (8, 6) \).
**Integration Approach:**
To find the area under the curve, calculate the integral of each segment:
- **Parabolic Segment (from \( x = -1 \) to \( x = 5 \)):** Calculate the area below the x-axis and subtract it from the area above the axis.
- **Linear Segment (from \( x = 5 \) to \( x = 8 \)):** Calculate the area of the triangle formed by the line and the x-axis.
Calculate these areas accurately using geometric shapes and add them to find \( \int_{-1}^{8} f(x) \, dx \).
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