8. Estimate the area under the graph of f (x) =÷ from x = 1 to x = 2 using %3D four approximating rectangles and left endpoints. Sketch the graph and the rectangles. Is your estimate an underestimate or an overestimate?
8. Estimate the area under the graph of f (x) =÷ from x = 1 to x = 2 using %3D four approximating rectangles and left endpoints. Sketch the graph and the rectangles. Is your estimate an underestimate or an overestimate?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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8. Estimate the area under the graph of
?(?) =1/? ???? ? = 1 ?? ? = 2 ?????
four approximating rectangles and left endpoints. Sketch the graph and the
rectangles. Is your estimate an underestimate or an overestimate?
![**Question 8:**
Estimate the area under the graph of \( f(x) = \frac{1}{x} \) from \( x = 1 \) to \( x = 2 \) using four approximating rectangles and left endpoints. Sketch the graph and the rectangles. Is your estimate an underestimate or an overestimate?
**Explanation:**
To solve this problem, you will calculate the area under the curve \( f(x) = \frac{1}{x} \) using a left Riemann sum with four rectangles. Follow these steps:
1. **Divide the Interval:**
- The interval from \( x = 1 \) to \( x = 2 \) is divided into four equal subintervals: \([1, 1.25]\), \([1.25, 1.5]\), \([1.5, 1.75]\), \([1.75, 2]\).
2. **Determine the Width of Each Rectangle:**
- The width of each rectangle is \(\Delta x = \frac{2 - 1}{4} = 0.25\).
3. **Identify Left Endpoints:**
- For each subinterval, the left endpoints are \(1.0\), \(1.25\), \(1.5\), and \(1.75\).
4. **Calculate the Height of Each Rectangle:**
- Evaluate the function at each left endpoint:
- \( f(1) = 1 \)
- \( f(1.25) = \frac{1}{1.25} = 0.8 \)
- \( f(1.5) = \frac{1}{1.5} \approx 0.67 \)
- \( f(1.75) = \frac{1}{1.75} \approx 0.57 \)
5. **Compute the Area of Each Rectangle:**
- Multiply the height by the width for each rectangle:
- Area 1: \( 1 \times 0.25 = 0.25\)
- Area 2: \( 0.8 \times 0.25 = 0.2\)
- Area 3: \( 0.67 \times 0.25 \approx 0.1675\)
- Area 4: \( 0.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6eaa4e2e-4872-4cc6-837d-08eedda6fd67%2Fef981ba7-161b-46ea-8581-d89f69973459%2Fqt6b621_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Question 8:**
Estimate the area under the graph of \( f(x) = \frac{1}{x} \) from \( x = 1 \) to \( x = 2 \) using four approximating rectangles and left endpoints. Sketch the graph and the rectangles. Is your estimate an underestimate or an overestimate?
**Explanation:**
To solve this problem, you will calculate the area under the curve \( f(x) = \frac{1}{x} \) using a left Riemann sum with four rectangles. Follow these steps:
1. **Divide the Interval:**
- The interval from \( x = 1 \) to \( x = 2 \) is divided into four equal subintervals: \([1, 1.25]\), \([1.25, 1.5]\), \([1.5, 1.75]\), \([1.75, 2]\).
2. **Determine the Width of Each Rectangle:**
- The width of each rectangle is \(\Delta x = \frac{2 - 1}{4} = 0.25\).
3. **Identify Left Endpoints:**
- For each subinterval, the left endpoints are \(1.0\), \(1.25\), \(1.5\), and \(1.75\).
4. **Calculate the Height of Each Rectangle:**
- Evaluate the function at each left endpoint:
- \( f(1) = 1 \)
- \( f(1.25) = \frac{1}{1.25} = 0.8 \)
- \( f(1.5) = \frac{1}{1.5} \approx 0.67 \)
- \( f(1.75) = \frac{1}{1.75} \approx 0.57 \)
5. **Compute the Area of Each Rectangle:**
- Multiply the height by the width for each rectangle:
- Area 1: \( 1 \times 0.25 = 0.25\)
- Area 2: \( 0.8 \times 0.25 = 0.2\)
- Area 3: \( 0.67 \times 0.25 \approx 0.1675\)
- Area 4: \( 0.
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