In this problem you will evaluate x dx using the 4 methods discussed in class. a) Estimate the value of fx dx using n= 4 subintervals and using the right endpoints as sample points. Draw the rectangles you used in this approximation. 5+ 4+ 3+ 2+ 0 b) Find the exact value of x dx using the Riemann sum definition with sample points being right endpoints n(n+1) 2 and the fact that Σί i=1 = c) Compute [x dx using the area interpretation. d) Computex dx using the FTC part 2 and the antiderivative (i.e. directly). 1
In this problem you will evaluate x dx using the 4 methods discussed in class. a) Estimate the value of fx dx using n= 4 subintervals and using the right endpoints as sample points. Draw the rectangles you used in this approximation. 5+ 4+ 3+ 2+ 0 b) Find the exact value of x dx using the Riemann sum definition with sample points being right endpoints n(n+1) 2 and the fact that Σί i=1 = c) Compute [x dx using the area interpretation. d) Computex dx using the FTC part 2 and the antiderivative (i.e. directly). 1
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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![## Evaluating the Integral ∫ from 1 to 5 of x dx
In this problem, you will evaluate the integral \(\int_{1}^{5} x \, dx\) using four different methods discussed in class.
### a) Estimation Using Right Endpoints
- **Task:** Estimate the value of \(\int_{1}^{5} x \, dx\) using \(n = 4\) subintervals and the right endpoints as sample points.
- **Instructions:** Draw the rectangles you used in this approximation.
**Graph Explanation:**
The graph provided displays the function \(f(x) = x\) from \(x = 1\) to \(x = 5\). To approximate the integral, divide the interval [1, 5] into 4 subintervals of equal width (each of width = 1). Using right endpoints, the heights of the rectangles will be the function values at \(x = 2, 3, 4, \text{and} 5\). Draw rectangles at these points:
1. Rectangle 1: Base from 1 to 2, height of 2.
2. Rectangle 2: Base from 2 to 3, height of 3.
3. Rectangle 3: Base from 3 to 4, height of 4.
4. Rectangle 4: Base from 4 to 5, height of 5.
### b) Exact Value Using Riemann Sum
- **Task:** Find the exact value of \(\int_{1}^{5} x \, dx\) using the Riemann sum definition with sample points being right endpoints. Use the fact that:
\[
\sum_{i=1}^{n} i = \frac{n(n+1)}{2}
\]
### c) Computing the Integral Using Area Interpretation
- **Task:** Compute \(\int_{1}^{5} x \, dx\) using the area interpretation.
### d) Computing the Integral Using the Fundamental Theorem of Calculus (FTC)
- **Task:** Compute \(\int_{1}^{5} x \, dx\) using FTC Part 2 and the antiderivative (i.e., directly).
Each of these methods will provide you with different approaches to evaluating the given integral, enhancing your understanding of integration techniques.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd65cec20-e50a-495e-bf9b-5a17b6c06d4e%2F5059b982-cf7d-4dcc-b542-3d73f6dc73f5%2Femhkug6_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Evaluating the Integral ∫ from 1 to 5 of x dx
In this problem, you will evaluate the integral \(\int_{1}^{5} x \, dx\) using four different methods discussed in class.
### a) Estimation Using Right Endpoints
- **Task:** Estimate the value of \(\int_{1}^{5} x \, dx\) using \(n = 4\) subintervals and the right endpoints as sample points.
- **Instructions:** Draw the rectangles you used in this approximation.
**Graph Explanation:**
The graph provided displays the function \(f(x) = x\) from \(x = 1\) to \(x = 5\). To approximate the integral, divide the interval [1, 5] into 4 subintervals of equal width (each of width = 1). Using right endpoints, the heights of the rectangles will be the function values at \(x = 2, 3, 4, \text{and} 5\). Draw rectangles at these points:
1. Rectangle 1: Base from 1 to 2, height of 2.
2. Rectangle 2: Base from 2 to 3, height of 3.
3. Rectangle 3: Base from 3 to 4, height of 4.
4. Rectangle 4: Base from 4 to 5, height of 5.
### b) Exact Value Using Riemann Sum
- **Task:** Find the exact value of \(\int_{1}^{5} x \, dx\) using the Riemann sum definition with sample points being right endpoints. Use the fact that:
\[
\sum_{i=1}^{n} i = \frac{n(n+1)}{2}
\]
### c) Computing the Integral Using Area Interpretation
- **Task:** Compute \(\int_{1}^{5} x \, dx\) using the area interpretation.
### d) Computing the Integral Using the Fundamental Theorem of Calculus (FTC)
- **Task:** Compute \(\int_{1}^{5} x \, dx\) using FTC Part 2 and the antiderivative (i.e., directly).
Each of these methods will provide you with different approaches to evaluating the given integral, enhancing your understanding of integration techniques.
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