Estimate the area under the graph of the function f (z) = Vz + 5 from z = -2 to z = 4 using a Riemann sum with n= 10 subintervals and midpoints. Round your answer to four decimal places. area- Number
Estimate the area under the graph of the function f (z) = Vz + 5 from z = -2 to z = 4 using a Riemann sum with n= 10 subintervals and midpoints. Round your answer to four decimal places. area- Number
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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Riemann Sum
Riemann Sums is a special type of approximation of the area under a curve by dividing it into multiple simple shapes like rectangles or trapezoids and is used in integrals when finite sums are involved. Figuring out the area of a curve is complex hence this method makes it simple. Usually, we take the help of different integration methods for this purpose. This is one of the major parts of integral calculus.
Riemann Integral
Bernhard Riemann's integral was the first systematic description of the integral of a function on an interval in the branch of mathematics known as real analysis.
Question
Estimate the area under the graph of the function of f(x) =
![### Estimating the Area Under a Curve Using Riemann Sums
In this practice problem, we aim to estimate the area under the graph of the function \( f(x) = \sqrt{x + 5} \) from \( x = -2 \) to \( x = 4 \). To achieve this, we will use a Riemann sum with \( n = 10 \) subintervals and midpoints.
The steps involved are as follows:
1. **Divide the interval \([-2, 4]\) into 10 equal subintervals:**
- The width of each subinterval \(\Delta x\) is calculated as:
\[
\Delta x = \frac{4 - (-2)}{10} = \frac{6}{10} = 0.6
\]
2. **Determine the midpoint of each subinterval:**
- The midpoint of the \(i\)-th subinterval \([x_{i-1}, x_i]\) is given by:
\[
\text{Midpoint} = \frac{x_{i-1} + x_i}{2}
\]
- Calculate the midpoints for each subinterval.
3. **Calculate \( f(x) \) at each midpoint:**
- Substitute the midpoint of each subinterval into the function \( f(x) = \sqrt{x + 5} \).
4. **Sum the products of the function values and the subinterval width:**
- The estimated area \( A \) is given by:
\[
A \approx \sum_{i=1}^{10} f(\text{Midpoint}_i) \cdot \Delta x
\]
Finally, round your answer to four decimal places.
### Example Calculation:
If you proceed with the calculations as outlined, you will obtain an approximate value for the area. Enter this value in the box provided:
\[ \text{area} = \ \underline{\text{Number}} \]
Please note that accurate calculations and rounding are essential to ensure precision.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcecc0901-ec5b-491a-8e4c-23ad559e3c3b%2Fdfb1d74d-fa53-45a7-a4bb-5c3cea0184c3%2Fdy9z20d_processed.png&w=3840&q=75)
Transcribed Image Text:### Estimating the Area Under a Curve Using Riemann Sums
In this practice problem, we aim to estimate the area under the graph of the function \( f(x) = \sqrt{x + 5} \) from \( x = -2 \) to \( x = 4 \). To achieve this, we will use a Riemann sum with \( n = 10 \) subintervals and midpoints.
The steps involved are as follows:
1. **Divide the interval \([-2, 4]\) into 10 equal subintervals:**
- The width of each subinterval \(\Delta x\) is calculated as:
\[
\Delta x = \frac{4 - (-2)}{10} = \frac{6}{10} = 0.6
\]
2. **Determine the midpoint of each subinterval:**
- The midpoint of the \(i\)-th subinterval \([x_{i-1}, x_i]\) is given by:
\[
\text{Midpoint} = \frac{x_{i-1} + x_i}{2}
\]
- Calculate the midpoints for each subinterval.
3. **Calculate \( f(x) \) at each midpoint:**
- Substitute the midpoint of each subinterval into the function \( f(x) = \sqrt{x + 5} \).
4. **Sum the products of the function values and the subinterval width:**
- The estimated area \( A \) is given by:
\[
A \approx \sum_{i=1}^{10} f(\text{Midpoint}_i) \cdot \Delta x
\]
Finally, round your answer to four decimal places.
### Example Calculation:
If you proceed with the calculations as outlined, you will obtain an approximate value for the area. Enter this value in the box provided:
\[ \text{area} = \ \underline{\text{Number}} \]
Please note that accurate calculations and rounding are essential to ensure precision.
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