### Left Riemann Sum Calculation Using Sigma Notation **Problem Statement:** Use sigma notation to write the left Riemann sum for \( f(x) = \sqrt{x^3 - 50} \) on the interval \([4, 12]\) with \( n = 60 \). Then evaluate the Riemann sum using a calculator. Round your answer to the nearest whole number. **Solution:** To calculate the left Riemann sum, follow these steps: 1. **Determine \(\Delta x\):** \[ \Delta x = \frac{b - a}{n} = \frac{12 - 4}{60} = \frac{8}{60} = \frac{2}{15} \] 2. **Set up the left Riemann sum in sigma notation:** \[ \text{Left Riemann Sum} = \sum_{i=0}^{n-1} f(a + i\Delta x) \Delta x \] For our function and interval: \[ \text{Left Riemann Sum} = \sum_{i=0}^{59} \sqrt{(4 + i \cdot \frac{2}{15})^3 - 50} \cdot \frac{2}{15} \] 3. **Evaluate the sum using a calculator:** Plug in the values and calculate the sum. After computation, round the result to the nearest whole number. **Example Calculation for \( i = 0 \) to understand the process:** For \( i = 0 \): \[ f(4 + 0 \cdot \frac{2}{15}) = \sqrt{4^3 - 50} = \sqrt{64 - 50} = \sqrt{14} \] So, one term of the sum is: \[ \sqrt{14} \cdot \frac{2}{15} \] Sum all 60 such terms and round the final result to the nearest whole number. **Note:** The complete computation typically involves using a calculator programmed to handle sigma sums or a software tool designed for numerical integration due to the complexity and large number of terms.
### Left Riemann Sum Calculation Using Sigma Notation **Problem Statement:** Use sigma notation to write the left Riemann sum for \( f(x) = \sqrt{x^3 - 50} \) on the interval \([4, 12]\) with \( n = 60 \). Then evaluate the Riemann sum using a calculator. Round your answer to the nearest whole number. **Solution:** To calculate the left Riemann sum, follow these steps: 1. **Determine \(\Delta x\):** \[ \Delta x = \frac{b - a}{n} = \frac{12 - 4}{60} = \frac{8}{60} = \frac{2}{15} \] 2. **Set up the left Riemann sum in sigma notation:** \[ \text{Left Riemann Sum} = \sum_{i=0}^{n-1} f(a + i\Delta x) \Delta x \] For our function and interval: \[ \text{Left Riemann Sum} = \sum_{i=0}^{59} \sqrt{(4 + i \cdot \frac{2}{15})^3 - 50} \cdot \frac{2}{15} \] 3. **Evaluate the sum using a calculator:** Plug in the values and calculate the sum. After computation, round the result to the nearest whole number. **Example Calculation for \( i = 0 \) to understand the process:** For \( i = 0 \): \[ f(4 + 0 \cdot \frac{2}{15}) = \sqrt{4^3 - 50} = \sqrt{64 - 50} = \sqrt{14} \] So, one term of the sum is: \[ \sqrt{14} \cdot \frac{2}{15} \] Sum all 60 such terms and round the final result to the nearest whole number. **Note:** The complete computation typically involves using a calculator programmed to handle sigma sums or a software tool designed for numerical integration due to the complexity and large number of terms.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Left Riemann Sum Calculation Using Sigma Notation
**Problem Statement:**
Use sigma notation to write the left Riemann sum for \( f(x) = \sqrt{x^3 - 50} \) on the interval \([4, 12]\) with \( n = 60 \). Then evaluate the Riemann sum using a calculator. Round your answer to the nearest whole number.
**Solution:**
To calculate the left Riemann sum, follow these steps:
1. **Determine \(\Delta x\):**
\[
\Delta x = \frac{b - a}{n} = \frac{12 - 4}{60} = \frac{8}{60} = \frac{2}{15}
\]
2. **Set up the left Riemann sum in sigma notation:**
\[
\text{Left Riemann Sum} = \sum_{i=0}^{n-1} f(a + i\Delta x) \Delta x
\]
For our function and interval:
\[
\text{Left Riemann Sum} = \sum_{i=0}^{59} \sqrt{(4 + i \cdot \frac{2}{15})^3 - 50} \cdot \frac{2}{15}
\]
3. **Evaluate the sum using a calculator:**
Plug in the values and calculate the sum. After computation, round the result to the nearest whole number.
**Example Calculation for \( i = 0 \) to understand the process:**
For \( i = 0 \):
\[
f(4 + 0 \cdot \frac{2}{15}) = \sqrt{4^3 - 50} = \sqrt{64 - 50} = \sqrt{14}
\]
So, one term of the sum is:
\[
\sqrt{14} \cdot \frac{2}{15}
\]
Sum all 60 such terms and round the final result to the nearest whole number.
**Note:**
The complete computation typically involves using a calculator programmed to handle sigma sums or a software tool designed for numerical integration due to the complexity and large number of terms.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F02ec4b82-5ff0-4052-a25d-a8fd5708f2e2%2Ffa7039d3-c017-4a3a-86a4-5dca1eed73da%2F9ztk8a_processed.png&w=3840&q=75)
Transcribed Image Text:### Left Riemann Sum Calculation Using Sigma Notation
**Problem Statement:**
Use sigma notation to write the left Riemann sum for \( f(x) = \sqrt{x^3 - 50} \) on the interval \([4, 12]\) with \( n = 60 \). Then evaluate the Riemann sum using a calculator. Round your answer to the nearest whole number.
**Solution:**
To calculate the left Riemann sum, follow these steps:
1. **Determine \(\Delta x\):**
\[
\Delta x = \frac{b - a}{n} = \frac{12 - 4}{60} = \frac{8}{60} = \frac{2}{15}
\]
2. **Set up the left Riemann sum in sigma notation:**
\[
\text{Left Riemann Sum} = \sum_{i=0}^{n-1} f(a + i\Delta x) \Delta x
\]
For our function and interval:
\[
\text{Left Riemann Sum} = \sum_{i=0}^{59} \sqrt{(4 + i \cdot \frac{2}{15})^3 - 50} \cdot \frac{2}{15}
\]
3. **Evaluate the sum using a calculator:**
Plug in the values and calculate the sum. After computation, round the result to the nearest whole number.
**Example Calculation for \( i = 0 \) to understand the process:**
For \( i = 0 \):
\[
f(4 + 0 \cdot \frac{2}{15}) = \sqrt{4^3 - 50} = \sqrt{64 - 50} = \sqrt{14}
\]
So, one term of the sum is:
\[
\sqrt{14} \cdot \frac{2}{15}
\]
Sum all 60 such terms and round the final result to the nearest whole number.
**Note:**
The complete computation typically involves using a calculator programmed to handle sigma sums or a software tool designed for numerical integration due to the complexity and large number of terms.
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