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
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Chapter1: Functions And Models
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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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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.
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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