### Evaluating Definite Integrals from a Graph #### Problem Statement Given the function graphed below, evaluate the definite integrals: \[ \int_{0}^{3} f(x) \, dx = \] \[ \int_{0}^{9} f(x) \, dx = \] #### Graph Description The graph shows a piecewise linear function with the following key points: - The function starts at point (0, -1). - It increases linearly to point (1, 2). - It then decreases linearly to point (3, 0). - The function remains constant from point (3, 0) to point (9, 0). #### Instructions 1. **Analyze Each Section:** - For \(0 \leq x \leq 1\), calculate the area of the triangle formed. - For \(1 \leq x \leq 3\), calculate the area of the triangle formed. - For \(3 \leq x \leq 9\), calculate the area of the rectangle (which is zero). 2. **Integrate Over Each Interval:** - Compute the definite integral for \([0, 3]\). - Compute the definite integral for \([0, 9]\). 3. **Discussions:** - Explain how to use the area under the curve to find the definite integrals. - Discuss any properties of definite integrals that help in simplifying calculations. By understanding the graphical representation and calculating the areas under the curve, you can accurately evaluate the definite integrals as required. Remember, the integral represents the net area, considering both above and below the x-axis contributions.
### Evaluating Definite Integrals from a Graph #### Problem Statement Given the function graphed below, evaluate the definite integrals: \[ \int_{0}^{3} f(x) \, dx = \] \[ \int_{0}^{9} f(x) \, dx = \] #### Graph Description The graph shows a piecewise linear function with the following key points: - The function starts at point (0, -1). - It increases linearly to point (1, 2). - It then decreases linearly to point (3, 0). - The function remains constant from point (3, 0) to point (9, 0). #### Instructions 1. **Analyze Each Section:** - For \(0 \leq x \leq 1\), calculate the area of the triangle formed. - For \(1 \leq x \leq 3\), calculate the area of the triangle formed. - For \(3 \leq x \leq 9\), calculate the area of the rectangle (which is zero). 2. **Integrate Over Each Interval:** - Compute the definite integral for \([0, 3]\). - Compute the definite integral for \([0, 9]\). 3. **Discussions:** - Explain how to use the area under the curve to find the definite integrals. - Discuss any properties of definite integrals that help in simplifying calculations. By understanding the graphical representation and calculating the areas under the curve, you can accurately evaluate the definite integrals as required. Remember, the integral represents the net area, considering both above and below the x-axis contributions.
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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![### Evaluating Definite Integrals from a Graph
#### Problem Statement
Given the function graphed below, evaluate the definite integrals:
\[ \int_{0}^{3} f(x) \, dx = \]
\[ \int_{0}^{9} f(x) \, dx = \]
#### Graph Description
The graph shows a piecewise linear function with the following key points:
- The function starts at point (0, -1).
- It increases linearly to point (1, 2).
- It then decreases linearly to point (3, 0).
- The function remains constant from point (3, 0) to point (9, 0).
#### Instructions
1. **Analyze Each Section:**
- For \(0 \leq x \leq 1\), calculate the area of the triangle formed.
- For \(1 \leq x \leq 3\), calculate the area of the triangle formed.
- For \(3 \leq x \leq 9\), calculate the area of the rectangle (which is zero).
2. **Integrate Over Each Interval:**
- Compute the definite integral for \([0, 3]\).
- Compute the definite integral for \([0, 9]\).
3. **Discussions:**
- Explain how to use the area under the curve to find the definite integrals.
- Discuss any properties of definite integrals that help in simplifying calculations.
By understanding the graphical representation and calculating the areas under the curve, you can accurately evaluate the definite integrals as required. Remember, the integral represents the net area, considering both above and below the x-axis contributions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa3a554af-d3bd-472f-bfdf-28ef9e412b08%2F2c3205ee-54de-413d-a37c-9aacc1a4137d%2Fxq0967w.jpeg&w=3840&q=75)
Transcribed Image Text:### Evaluating Definite Integrals from a Graph
#### Problem Statement
Given the function graphed below, evaluate the definite integrals:
\[ \int_{0}^{3} f(x) \, dx = \]
\[ \int_{0}^{9} f(x) \, dx = \]
#### Graph Description
The graph shows a piecewise linear function with the following key points:
- The function starts at point (0, -1).
- It increases linearly to point (1, 2).
- It then decreases linearly to point (3, 0).
- The function remains constant from point (3, 0) to point (9, 0).
#### Instructions
1. **Analyze Each Section:**
- For \(0 \leq x \leq 1\), calculate the area of the triangle formed.
- For \(1 \leq x \leq 3\), calculate the area of the triangle formed.
- For \(3 \leq x \leq 9\), calculate the area of the rectangle (which is zero).
2. **Integrate Over Each Interval:**
- Compute the definite integral for \([0, 3]\).
- Compute the definite integral for \([0, 9]\).
3. **Discussions:**
- Explain how to use the area under the curve to find the definite integrals.
- Discuss any properties of definite integrals that help in simplifying calculations.
By understanding the graphical representation and calculating the areas under the curve, you can accurately evaluate the definite integrals as required. Remember, the integral represents the net area, considering both above and below the x-axis contributions.
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