A) estimate the area of the graph of f(x) = 25 - x² From x=0 to x=5 using approximating rectangles and right end points estimate = B) Repeat using left endpoints estimate= C) Repeat using midpoints estimate =
A) estimate the area of the graph of f(x) = 25 - x² From x=0 to x=5 using approximating rectangles and right end points estimate = B) Repeat using left endpoints estimate= C) Repeat using midpoints estimate =
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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![**Title: Estimating the Area Under a Curve Using Rectangular Approximation**
**Objective:**
Learn how to estimate the area under the graph of a function using different methods of rectangular approximation — right endpoints, left endpoints, and midpoints.
**Problem Statement:**
Given the function \( f(x) = 25 - x^2 \), estimate the area under the curve from \( x = 0 \) to \( x = 5 \) using 5 approximating rectangles and the following methods:
**A) Using Right Endpoints:**
Estimate the area of the graph of \( f(x) = 25 - x^2 \) from \( x = 0 \) to \( x = 5 \) using 5 approximating rectangles and right endpoints.
- **Estimate:**
**B) Using Left Endpoints:**
Repeat the process using left endpoints.
- **Estimate:**
**C) Using Midpoints:**
Repeat the process using midpoints.
- **Estimate:**
**Instructions for Calculations:**
1. **Divide the interval** \([0, 5]\) into 5 equally spaced sub-intervals. Each sub-interval will have a width \( \Delta x = \frac{5-0}{5} = 1 \).
2. **Calculate the height of each rectangle** based on the evaluation of the function \( f(x) \) at the specified points (right endpoints, left endpoints, or midpoints).
- **For right endpoints:** Evaluate \( f(x) \) at \( x = 1, 2, 3, 4, 5 \).
- **For left endpoints:** Evaluate \( f(x) \) at \( x = 0, 1, 2, 3, 4 \).
- **For midpoints:** Evaluate \( f(x) \) at \( x = 0.5, 1.5, 2.5, 3.5, 4.5 \).
3. **Multiply the height by the width** \( (\Delta x) \) to find the area of each rectangle.
4. **Sum the areas** of the 5 rectangles to get the total estimated area under the curve.
This method of approximation is a foundational concept in integral calculus and helps in understanding how definite integrals work.
Practicing these manual calculations strengthens your grasp of the integral's geometric interpretation and aids in visualization](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9d7d1679-9649-4c1d-b706-e07713ee238e%2F61f5d0e4-b192-47c5-86ea-25cf31c71d8e%2F4qeapa9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Estimating the Area Under a Curve Using Rectangular Approximation**
**Objective:**
Learn how to estimate the area under the graph of a function using different methods of rectangular approximation — right endpoints, left endpoints, and midpoints.
**Problem Statement:**
Given the function \( f(x) = 25 - x^2 \), estimate the area under the curve from \( x = 0 \) to \( x = 5 \) using 5 approximating rectangles and the following methods:
**A) Using Right Endpoints:**
Estimate the area of the graph of \( f(x) = 25 - x^2 \) from \( x = 0 \) to \( x = 5 \) using 5 approximating rectangles and right endpoints.
- **Estimate:**
**B) Using Left Endpoints:**
Repeat the process using left endpoints.
- **Estimate:**
**C) Using Midpoints:**
Repeat the process using midpoints.
- **Estimate:**
**Instructions for Calculations:**
1. **Divide the interval** \([0, 5]\) into 5 equally spaced sub-intervals. Each sub-interval will have a width \( \Delta x = \frac{5-0}{5} = 1 \).
2. **Calculate the height of each rectangle** based on the evaluation of the function \( f(x) \) at the specified points (right endpoints, left endpoints, or midpoints).
- **For right endpoints:** Evaluate \( f(x) \) at \( x = 1, 2, 3, 4, 5 \).
- **For left endpoints:** Evaluate \( f(x) \) at \( x = 0, 1, 2, 3, 4 \).
- **For midpoints:** Evaluate \( f(x) \) at \( x = 0.5, 1.5, 2.5, 3.5, 4.5 \).
3. **Multiply the height by the width** \( (\Delta x) \) to find the area of each rectangle.
4. **Sum the areas** of the 5 rectangles to get the total estimated area under the curve.
This method of approximation is a foundational concept in integral calculus and helps in understanding how definite integrals work.
Practicing these manual calculations strengthens your grasp of the integral's geometric interpretation and aids in visualization
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