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:**
Express \(\int_1^4 x^3 \, dx\) as a limit of the Riemann Sums taking sample points to be right endpoints.
---
**Explanation:**
To express the definite integral \(\int_1^4 x^3 \, dx\) as a limit of Riemann Sums using right endpoints, follow these steps:
1. **Partition the Interval:**
- Divide the interval \([1, 4]\) into \(n\) subintervals of equal width, \(\Delta x = \frac{4-1}{n} = \frac{3}{n}\).
2. **Right Endpoints:**
- For each subinterval \([x_{i-1}, x_i]\), the right endpoint will be \(x_i = 1 + i\Delta x = 1 + \frac{3i}{n}\).
3. **Riemann Sum Expression:**
- The Riemann sum using right endpoints is given by:
\[
S_n = \sum_{i=1}^{n} f(x_i) \Delta x = \sum_{i=1}^{n} \left(1 + \frac{3i}{n}\right)^3 \frac{3}{n}
\]
4. **Limit of the Riemann Sums:**
- The integral is the limit of these sums as \(n \to \infty\):
\[
\int_1^4 x^3 \, dx = \lim_{n \to \infty} S_n = \lim_{n \to \infty} \sum_{i=1}^{n} \left(1 + \frac{3i}{n}\right)^3 \frac{3}{n}
\]
This method allows the calculation of the integral by approximating the area under the curve \(f(x) = x^3\) over the interval \([1, 4]\) using Riemann sums.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa4b64424-2f1e-400b-a2f3-ccc24d7d74e8%2F3614c6bf-f010-4f50-a2e6-1574e7eb8d2f%2Fes9sblv_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Express \(\int_1^4 x^3 \, dx\) as a limit of the Riemann Sums taking sample points to be right endpoints.
---
**Explanation:**
To express the definite integral \(\int_1^4 x^3 \, dx\) as a limit of Riemann Sums using right endpoints, follow these steps:
1. **Partition the Interval:**
- Divide the interval \([1, 4]\) into \(n\) subintervals of equal width, \(\Delta x = \frac{4-1}{n} = \frac{3}{n}\).
2. **Right Endpoints:**
- For each subinterval \([x_{i-1}, x_i]\), the right endpoint will be \(x_i = 1 + i\Delta x = 1 + \frac{3i}{n}\).
3. **Riemann Sum Expression:**
- The Riemann sum using right endpoints is given by:
\[
S_n = \sum_{i=1}^{n} f(x_i) \Delta x = \sum_{i=1}^{n} \left(1 + \frac{3i}{n}\right)^3 \frac{3}{n}
\]
4. **Limit of the Riemann Sums:**
- The integral is the limit of these sums as \(n \to \infty\):
\[
\int_1^4 x^3 \, dx = \lim_{n \to \infty} S_n = \lim_{n \to \infty} \sum_{i=1}^{n} \left(1 + \frac{3i}{n}\right)^3 \frac{3}{n}
\]
This method allows the calculation of the integral by approximating the area under the curve \(f(x) = x^3\) over the interval \([1, 4]\) using Riemann sums.
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