
Homework:HW SECTION 8.8
Question 4, 8.8.45
Part 1 of 5
HW Score: 37.5%, 3 of 8 points
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Part 1
Consider the integral Integral from 0 to 1 e Superscript 6 x Baseline dx with nequals25.
a. Find the trapezoid rule approximations to the integral using n and 2n subintervals.
b. Find the​ Simpson's rule approximation to the integral using 2n subintervals.
c. Compute the absolute errors in the trapezoid rule and​ Simpson's rule with 2n subintervals.
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Part 1
a. What is the trapezoid approximation with n​ subintervals?
Upper T left parenthesis 25 right parenthesisequals
  
enter your response here ​(Round to six decimal places as​ needed.)
Upper T left parenthesis 20 right parenthesis(Round to six decimal places as needed.)Upper T left parenthesis 40 right parenthesis(Round to six decimal places as needed.)Upper S left parenthesis 40 right parenthesis(Round to six decimal places as needed.)What is the error in the trapezoid rule approximation with 2n subintervals?(Use scientific notation. Round to three decimal places as needed. Use the multiplication symbol in the math palette as needed.)What is the error in the Simpson's rule approximation with 2n subintervals?(Use scientific notation. Round to three decimal places as needed. Use the multiplication symbol in the math palette as needed.)Upper T left parenthesis 25 right parenthesis(Round to six decimal places as needed.)
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Let's break down the problem step by step:

### Problem Statement:
We need to approximate the integral:

\[
\int_0^1 e^{6x} \, dx
\]

using the **Trapezoid Rule** and **Simpson's Rule** for \( n = 25 \) subintervals and \( 2n = 50 \) subintervals.

### Step 1: Trapezoid Rule Approximation

The Trapezoid Rule formula for \( n \) subintervals is:

\[
T_n = \frac{b - a}{2n} \left( f(a) + 2 \sum_{i=1}^{n-1} f(x_i) + f(b) \right)
\]

where \( a = 0 \), \( b = 1 \), and \( f(x) = e^{6x} \).

1. **Calculate for \( n = 25 \) subintervals:**

2. **Calculate for \( 2n = 50 \) subintervals:**

### Step 2: Simpson's Rule Approximation

Simpson's Rule formula for \( 2n \) subintervals is:

\[
S_{2n} = \frac{b - a}{6n} \left( f(a) + 4 \sum_{\text{odd } i} f(x_i) + 2 \sum_{\text{even } i} f(x_i) + f(b) \right)
\]

### Step 3: Calculate Errors

- **Error in Trapezoid Rule with \( 2n \) subintervals:**

The error for the Trapezoid Rule is calculated as:

\[
\text{Error} = \left| \text{Exact Value} - T_{2n} \right|
\]

- **Error in Simpson's Rule with \( 2n \) subintervals:**

The error for Simpson's Rule is calculated as:

\[
\text{Error} = \left| \text{Exact Value} - S_{2n} \right|
\]

### Exact Value of the Integral:
The exact value of the integral can be calculated as:

\[
\int_0^1 e^{6x} \, dx = \left[ \frac{e^{6x}}{6} \right]_0^1 = \frac{e^6 - 1}{6}
\]

Let's calculate each value step by step.

I'll start by calculating the exact integral, then proceed with the Trapezoid and Simpson approximations.Here are the results:

1. **Trapezoid Rule Approximation with \( n = 25 \) subintervals:**
   \[
   T(25) \approx 67.393100
   \]

2. **Trapezoid Rule Approximation with \( 2n = 50 \) subintervals:**
   \[
   T(50) \approx 67.151932
   \]

3. **Simpson's Rule Approximation with \( 2n = 50 \) subintervals:**
   \[
   S(50) \approx 67.071543
   \]

4. **Error in Trapezoid Rule Approximation with \( 2n = 50 \) subintervals:**
   \[
   \text{Error} \approx 0.080 \text{ (in scientific notation: } 8.047 \times 10^{-2} \text{)}
   \]

5. **Error in Simpson's Rule Approximation with \( 2n = 50 \) subintervals:**
   \[
   \text{Error} \approx 7.713 \times 10^{-5}
   \]

Would you like further details on any of these calculations, or do you have any additional questions?

Here are 5 related questions you might explore:
1. How does the error change if you increase the number of subintervals further?
2. What is the general formula for the error in the Trapezoid Rule?
3. How does Simpson's Rule compare to the Trapezoid Rule in terms of accuracy?
4. Can you derive the exact value of the integral manually using integration techniques?
5. What happens if you apply the Midpoint Rule instead?

**Tip:** Simpson's Rule is usually more accurate than the Trapezoid Rule for the same number of subintervals, especially for functions with smooth curves.

Consider the integral Integral from 0 to 1 e Superscript 6 x Baseline dx with nequals25. a. Find the trapezoid rule approximations to the integral using n and 2n subintervals. b. Find the​ Simpson's rule approximation to the integral using 2n subintervals. c. Compute the absolute errors in the trapezoid rule and​ Simpson's rule with 2n subintervals.

Explore numerical integration techniques by approximating the integral of e^(6x) from 0 to 1 using the Trapezoid and Simpson's Rules with 25 and 50 subintervals. Learn how to compute absolute errors and understand the accuracy of each method. Suitable for undergraduate Calculus students.

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