Homework Help is Here – Start Your Trial Now!
Math
Calculus
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
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...
See similar textbooks
icon
Related questions
Question
## 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.
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.
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 5 steps with 6 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Precalculus
Precalculus
Calculus
ISBN:
9780135189405
Author:
Michael Sullivan
Publisher:
PEARSON
Calculus: Early Transcendental Functions
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning
SEE MORE TEXTBOOKS