11. Consider the region bounded by f(x)=x² and the x-axis over the interval [0, 2]. Suppose the area of the region is to be approximated by n equal-width rectangles using the Right Endpoints to evaluate height. Match each expression with its role in the process. The width of each rectangle. The height of the ith rectangle at the right endpoint of the ith subinterval. The area of the ith rectangle. The sum of the areas of the n rectangles. 3 812 [A] [B] [C] [D] n 8 [n(n+1)(2n+1)] n³ 6
11. Consider the region bounded by f(x)=x² and the x-axis over the interval [0, 2]. Suppose the area of the region is to be approximated by n equal-width rectangles using the Right Endpoints to evaluate height. Match each expression with its role in the process. The width of each rectangle. The height of the ith rectangle at the right endpoint of the ith subinterval. The area of the ith rectangle. The sum of the areas of the n rectangles. 3 812 [A] [B] [C] [D] n 8 [n(n+1)(2n+1)] n³ 6
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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![**Transcription for Educational Website**
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**Problem 11:**
Consider the region bounded by \( f(x) = x^2 \) and the x-axis over the interval \([0, 2]\). Suppose the area of the region is to be approximated by \( n \) equal-width rectangles using the Right Endpoints to evaluate height. Match each expression with its role in the process.
1. **The width of each rectangle.**
2. **The height of the \( i^{th} \) rectangle at the right endpoint of the \( i^{th} \) subinterval.**
3. **The area of the \( i^{th} \) rectangle.**
- [A] \(\frac{8i^2}{n^3}\)
- [B] \(\frac{2}{n}\)
- [C] \(\frac{4i^2}{n^2}\)
- [D] \(\frac{8}{n^3} \left[ \frac{n(n+1)(2n+1)}{6} \right]\)
4. **The sum of the areas of the \( n \) rectangles.**
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**Graph/Diagram Explanation:**
The diagram on the right is a graph of the function \( f(x) = x^2 \) from 0 to 2. The curve is a part of a parabola that opens upwards. The right side of the parabola extends to 2 on the x-axis and the height of the curve increases as it moves towards this endpoint. This illustrates how the function \( x^2 \) shapes the region under consideration when forming the approximating rectangles.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5ef4066a-09c4-44d3-9899-dbf8cf9ac919%2F7c926603-4152-427e-9900-22315bb33d92%2F27kyd9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Transcription for Educational Website**
---
**Problem 11:**
Consider the region bounded by \( f(x) = x^2 \) and the x-axis over the interval \([0, 2]\). Suppose the area of the region is to be approximated by \( n \) equal-width rectangles using the Right Endpoints to evaluate height. Match each expression with its role in the process.
1. **The width of each rectangle.**
2. **The height of the \( i^{th} \) rectangle at the right endpoint of the \( i^{th} \) subinterval.**
3. **The area of the \( i^{th} \) rectangle.**
- [A] \(\frac{8i^2}{n^3}\)
- [B] \(\frac{2}{n}\)
- [C] \(\frac{4i^2}{n^2}\)
- [D] \(\frac{8}{n^3} \left[ \frac{n(n+1)(2n+1)}{6} \right]\)
4. **The sum of the areas of the \( n \) rectangles.**
---
**Graph/Diagram Explanation:**
The diagram on the right is a graph of the function \( f(x) = x^2 \) from 0 to 2. The curve is a part of a parabola that opens upwards. The right side of the parabola extends to 2 on the x-axis and the height of the curve increases as it moves towards this endpoint. This illustrates how the function \( x^2 \) shapes the region under consideration when forming the approximating rectangles.
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