Find the area of the region bounded by the three lines below. [Hint: You will have to split up the integral into two parts.] 7 33x, A = y = 5 x, y = -x, y=-3x + 16
Find the area of the region bounded by the three lines below. [Hint: You will have to split up the integral into two parts.] 7 33x, A = y = 5 x, y = -x, y=-3x + 16
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:**
Find the area of the region bounded by the three lines below. [Hint: You will have to split up the integral into two parts.]
\[ y = \frac{1}{5}x \]
\[ y = \frac{7}{3}x \]
\[ y = -3x + 16 \]
\( \text{A} = \ \) [Empty box for the answer]
**Detailed Explanation:**
This problem requires calculating the area of the region formed by the intersection of three lines, which is a calculus-based approach involving integration. Given the functions are linear, the area between them will form polygonal segments, which can be calculated by finding the definite integrals of the bounding functions and summing or subtracting as necessary.
1. **Intersection Points:**
- First, find the points of intersection between the lines to determine the limits for your integrals.
- Find the intersection of \( y = \frac{1}{5}x \) and \( y = \frac{7}{3}x \).
- Find the intersection of \( y = \frac{1}{5}x \) and \( y = -3x + 16 \).
- Find the intersection of \( y = \frac{7}{3}x \) and \( y = -3x + 16 \).
2. **Setting up Integrals:**
- Once you have the intersection points, split the calculation into two parts as suggested.
- Set up and evaluate the definite integrals between the appropriate limits.
- Example integrals (after determining limits \(a\) and \(b\), and other points):
\[
\int_{a}^{b} \left( \frac{7}{3}x - \frac{1}{5}x \right) \, dx \quad \text{and} \quad \int_{b}^{c} \left( (-3x + 16) - \frac{7}{3}x \right) \, dx
\]
3. **Calculate Area \( A \):**
- The total area \( A \) will be the sum of the two integrals representing the areas between curves.
Without solving the integrals here, the steps outline the approach to follow for using calculus to determine the bounded area. Enter the calculated value of the area in the provided box to complete the problem.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F43f97564-e82e-4b67-89b1-801d2ce81d48%2Fd597b798-45ad-4826-b1e8-69b9423a91a0%2F69f84k_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the area of the region bounded by the three lines below. [Hint: You will have to split up the integral into two parts.]
\[ y = \frac{1}{5}x \]
\[ y = \frac{7}{3}x \]
\[ y = -3x + 16 \]
\( \text{A} = \ \) [Empty box for the answer]
**Detailed Explanation:**
This problem requires calculating the area of the region formed by the intersection of three lines, which is a calculus-based approach involving integration. Given the functions are linear, the area between them will form polygonal segments, which can be calculated by finding the definite integrals of the bounding functions and summing or subtracting as necessary.
1. **Intersection Points:**
- First, find the points of intersection between the lines to determine the limits for your integrals.
- Find the intersection of \( y = \frac{1}{5}x \) and \( y = \frac{7}{3}x \).
- Find the intersection of \( y = \frac{1}{5}x \) and \( y = -3x + 16 \).
- Find the intersection of \( y = \frac{7}{3}x \) and \( y = -3x + 16 \).
2. **Setting up Integrals:**
- Once you have the intersection points, split the calculation into two parts as suggested.
- Set up and evaluate the definite integrals between the appropriate limits.
- Example integrals (after determining limits \(a\) and \(b\), and other points):
\[
\int_{a}^{b} \left( \frac{7}{3}x - \frac{1}{5}x \right) \, dx \quad \text{and} \quad \int_{b}^{c} \left( (-3x + 16) - \frac{7}{3}x \right) \, dx
\]
3. **Calculate Area \( A \):**
- The total area \( A \) will be the sum of the two integrals representing the areas between curves.
Without solving the integrals here, the steps outline the approach to follow for using calculus to determine the bounded area. Enter the calculated value of the area in the provided box to complete the problem.
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