Estimate the area under the graph of f(x) = x² +5 over the interval [0, 10] using 10 approximating rectangles and midpoints as sample points. M10 = Repeat the approximation using using 50 approximating rectangles. M50 = Repeat the approximation using using 100 approximating rectangles. M100 = Repeat the approximation using using 500 approximating rectangles. M500 = These consecutive approximations appear to be approaching what value? So, it appears that the following integral expression is valid: S dr =

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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### Estimating the Area Under a Curve

#### Problem Statement
Estimate the area under the graph of \( f(x) = x^2 + 5 \) over the interval \([0, 10]\) using approximating rectangles with midpoints as sample points.

#### Calculations

1. **Using 10 Approximating Rectangles:**
   \[
   M_{10} = \text{[Enter Calculation Here]}
   \]

2. **Using 50 Approximating Rectangles:**
   \[
   M_{50} = \text{[Enter Calculation Here]}
   \]

3. **Using 100 Approximating Rectangles:**
   \[
   M_{100} = \text{[Enter Calculation Here]}
   \]

4. **Using 500 Approximating Rectangles:**
   \[
   M_{500} = \text{[Enter Calculation Here]}
   \]

#### Observations

These consecutive approximations appear to be approaching what value?
\[
\text{[Approaching Value]}
\]

#### Integral Expression

So, it appears that the following integral expression is valid:
\[
\int_{0}^{10} (x^2 + 5) \, dx = \text{[Enter Exact Value]}
\]
Transcribed Image Text:### Estimating the Area Under a Curve #### Problem Statement Estimate the area under the graph of \( f(x) = x^2 + 5 \) over the interval \([0, 10]\) using approximating rectangles with midpoints as sample points. #### Calculations 1. **Using 10 Approximating Rectangles:** \[ M_{10} = \text{[Enter Calculation Here]} \] 2. **Using 50 Approximating Rectangles:** \[ M_{50} = \text{[Enter Calculation Here]} \] 3. **Using 100 Approximating Rectangles:** \[ M_{100} = \text{[Enter Calculation Here]} \] 4. **Using 500 Approximating Rectangles:** \[ M_{500} = \text{[Enter Calculation Here]} \] #### Observations These consecutive approximations appear to be approaching what value? \[ \text{[Approaching Value]} \] #### Integral Expression So, it appears that the following integral expression is valid: \[ \int_{0}^{10} (x^2 + 5) \, dx = \text{[Enter Exact Value]} \]
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