3. Given f(x)dx integrals. You show the definite integral properties for these evaluations. 4 and f(x)dx -1 determine the values of the following definite %3D (a) So f(x)dr (b) f(x)dx (c) 4f(x)dx
3. Given f(x)dx integrals. You show the definite integral properties for these evaluations. 4 and f(x)dx -1 determine the values of the following definite %3D (a) So f(x)dr (b) f(x)dx (c) 4f(x)dx
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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![**Problem Statement:**
Given the definite integrals:
\[
\int_{0}^{3} f(x) \, dx = 4 \quad \text{and} \quad \int_{3}^{6} f(x) \, dx = -1
\]
Determine the values of the following definite integrals. Use the definite integral properties for these evaluations.
**Questions:**
(a) \(\int_{0}^{6} f(x) \, dx\)
(b) \(\int_{3}^{0} f(x) \, dx\)
(c) \(\int_{3}^{6} 4f(x) \, dx\)
**Solution Approach:**
To solve these problems, we will use the properties of definite integrals:
- **Additivity Property:** If \(\int_{a}^{b} f(x) \, dx = A\) and \(\int_{b}^{c} f(x) \, dx = B\), then \(\int_{a}^{c} f(x) \, dx = A + B\).
- **Reversal of Limits:** \(\int_{b}^{a} f(x) \, dx = - \int_{a}^{b} f(x) \, dx\).
- **Scaling Rule:** \(\int_{a}^{b} c \cdot f(x) \, dx = c \cdot \int_{a}^{b} f(x) \, dx\), where \(c\) is a constant.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbd5edf4c-1bea-43aa-8c8d-f5f43a4d0b49%2F82518b9e-db39-4c1f-8c57-5d649cd661ac%2Fhmugqgb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Given the definite integrals:
\[
\int_{0}^{3} f(x) \, dx = 4 \quad \text{and} \quad \int_{3}^{6} f(x) \, dx = -1
\]
Determine the values of the following definite integrals. Use the definite integral properties for these evaluations.
**Questions:**
(a) \(\int_{0}^{6} f(x) \, dx\)
(b) \(\int_{3}^{0} f(x) \, dx\)
(c) \(\int_{3}^{6} 4f(x) \, dx\)
**Solution Approach:**
To solve these problems, we will use the properties of definite integrals:
- **Additivity Property:** If \(\int_{a}^{b} f(x) \, dx = A\) and \(\int_{b}^{c} f(x) \, dx = B\), then \(\int_{a}^{c} f(x) \, dx = A + B\).
- **Reversal of Limits:** \(\int_{b}^{a} f(x) \, dx = - \int_{a}^{b} f(x) \, dx\).
- **Scaling Rule:** \(\int_{a}^{b} c \cdot f(x) \, dx = c \cdot \int_{a}^{b} f(x) \, dx\), where \(c\) is a constant.
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