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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![**Estimating the Area Under a Curve Using Rectangular Approximation**
In this example, we will estimate the area under the graph of the function \( f(x) = \frac{1}{x + 2} \) over the interval \([0, 3]\). We will use eight approximation rectangles with right endpoints.
**Procedure:**
1. **Divide the Interval:** The interval \([0, 3]\) is divided into eight equal subintervals. The width (\(\Delta x\)) of each rectangle is \(\frac{3 - 0}{8} = 0.375\).
2. **Determine Right Endpoints:** For each subinterval, use the right endpoint to determine the height of the rectangle. The endpoints are calculated as follows:
- \(x_1 = 0 + 0.375 \times 1\)
- \(x_2 = 0 + 0.375 \times 2\)
- \(x_3 = 0 + 0.375 \times 3\)
- ...
- \(x_8 = 0 + 0.375 \times 8\)
3. **Calculate Rectangle Areas:** For each rectangle, calculate the area using the formula: \(\text{Area} = f(x_i) \times \Delta x\), where \(f(x_i) = \frac{1}{x_i + 2}\).
4. **Sum the Areas:** The total approximate area \(R_n\) is the sum of the areas of all rectangles.
**Formulas:**
- **Right Endpoint Approximation:**
\[ R_n = \sum_{i=1}^{n} f(x_i) \times \Delta x \]
- **Left Endpoint Approximation:** Although not required for this problem, the left endpoint approximation would use the starting points of the subintervals.
**Symbols:**
- \(R_n\): Sum using right endpoints
- \(L_n\): Sum using left endpoints
Fill in the calculations to find the approximations:
- \( R_n = \) [Insert the calculated sum using right endpoints]
- \( L_n = \) [Not calculated in this example]
This method of Riemann sums helps provide an estimate for the definite integral of the function over the specified interval.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fee8bddee-32fd-433e-8085-8fa50d25b5ad%2F754b8b1d-3d4f-458f-a017-7384a53b6f13%2F2jiggbx_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Estimating the Area Under a Curve Using Rectangular Approximation**
In this example, we will estimate the area under the graph of the function \( f(x) = \frac{1}{x + 2} \) over the interval \([0, 3]\). We will use eight approximation rectangles with right endpoints.
**Procedure:**
1. **Divide the Interval:** The interval \([0, 3]\) is divided into eight equal subintervals. The width (\(\Delta x\)) of each rectangle is \(\frac{3 - 0}{8} = 0.375\).
2. **Determine Right Endpoints:** For each subinterval, use the right endpoint to determine the height of the rectangle. The endpoints are calculated as follows:
- \(x_1 = 0 + 0.375 \times 1\)
- \(x_2 = 0 + 0.375 \times 2\)
- \(x_3 = 0 + 0.375 \times 3\)
- ...
- \(x_8 = 0 + 0.375 \times 8\)
3. **Calculate Rectangle Areas:** For each rectangle, calculate the area using the formula: \(\text{Area} = f(x_i) \times \Delta x\), where \(f(x_i) = \frac{1}{x_i + 2}\).
4. **Sum the Areas:** The total approximate area \(R_n\) is the sum of the areas of all rectangles.
**Formulas:**
- **Right Endpoint Approximation:**
\[ R_n = \sum_{i=1}^{n} f(x_i) \times \Delta x \]
- **Left Endpoint Approximation:** Although not required for this problem, the left endpoint approximation would use the starting points of the subintervals.
**Symbols:**
- \(R_n\): Sum using right endpoints
- \(L_n\): Sum using left endpoints
Fill in the calculations to find the approximations:
- \( R_n = \) [Insert the calculated sum using right endpoints]
- \( L_n = \) [Not calculated in this example]
This method of Riemann sums helps provide an estimate for the definite integral of the function over the specified interval.
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