(a)
Find the approximations T4 and M4 for the following integral. (Round your answers to six decimal places.)
2

1
31e1⁄x dx
T4
=	  
Correct: Your answer is correct.
M4
=	  
Correct: Your answer is correct.
(b)
Estimate the errors in the approximations of part (a) using the smallest possible value for K according to the theorem about error bounds for trapezoidal and midpoint rules. (Round your answers to six decimal places.)
|ET|
≤	  
Correct: Your answer is correct.
|EM|
≤	  
Correct: Your answer is correct.
(c)
Using the values of K from part (b), how large do we have to choose n so that the approximations 
Tn
 and 
Mn
 to the integral in part (a) are accurate to within 0.0001?

To approach the problem, we will break it down into parts (a), (b), and (c), and follow a structured method to solve each.

### (a) Approximating the integral using the Trapezoidal Rule (\(T_4\)) and Midpoint Rule (\(M_4\)):

The integral given is:

\[
I = \int_1^2 \frac{31e^{1/x}}{x} \, dx
\]

We approximate this using \( n = 4 \) subintervals.

#### Trapezoidal Rule (\(T_4\)):
The trapezoidal rule formula for \( n \) subintervals is:

\[
T_n = \frac{b - a}{2n} \left[ f(x_0) + 2f(x_1) + 2f(x_2) + \cdots + 2f(x_{n-1}) + f(x_n) \right]
\]

For \( n = 4 \), the width of each subinterval is \( h = \frac{2 - 1}{4} = 0.25 \). The points \( x_0, x_1, x_2, x_3, x_4 \) are:

\[
x_0 = 1, \, x_1 = 1.25, \, x_2 = 1.5, \, x_3 = 1.75, \, x_4 = 2
\]

Substitute these into the formula for \(T_4\), compute the function values at each point, and sum them to get the approximation \(T_4\).

#### Midpoint Rule (\(M_4\)):
The midpoint rule formula for \( n \) subintervals is:

\[
M_n = \frac{b - a}{n} \left[ f \left( \frac{x_0 + x_1}{2} \right) + f \left( \frac{x_1 + x_2}{2} \right) + \cdots + f \left( \frac{x_{n-1} + x_n}{2} \right) \right]
\]

For \( n = 4 \), the midpoints are:

\[
m_1 = 1.125, \, m_2 = 1.375, \, m_3 = 1.625, \, m_4 = 1.875
\]

Substitute these values into the formula for \( M_4 \), compute the function values at the midpoints, and sum them to get the approximation \(M_4\).

### (b) Estimating errors using the error bounds for Trapezoidal and Midpoint Rules:

To estimate the errors, we use the following error bounds for the Trapezoidal and Midpoint rules:

\[
|E_T| \leq \frac{K(b-a)^3}{12n^2}
\]
\[
|E_M| \leq \frac{K(b-a)^3}{24n^2}
\]

Where \( K \) is an upper bound for \( |f''(x)| \) on the interval \( [a, b] \).

#### Step 1: Find \( f(x) \) and \( f''(x) \):

Given that \( f(x) = \frac{31e^{1/x}}{x} \), we need to compute its second derivative \( f''(x) \) and find the maximum value of \( |f''(x)| \) on \( [1, 2] \).

#### Step 2: Find \( K \) (the smallest possible value):

Compute \( |f''(x)| \) and find the maximum value on the interval \( [1, 2] \). This will give us the value of \( K \).

#### Step 3: Plug values into error bounds:

Using the value of \( K \), substitute into the error formulas for \( E_T \) and \( E_M \) to estimate the errors in the approximations \( T_4 \) and \( M_4 \).

### (c) Determining the required \( n \) for desired accuracy:

We are asked to find how large \( n \) must be for the approximations \( T_n \) and \( M_n \) to be accurate within 0.0001. We use the error bounds from part (b):

\[
\frac{K(b-a)^3}{12n^2} \leq 0.0001
\]
\[
\frac{K(b-a)^3}{24n^2} \leq 0.0001
\]

Solve for \( n \) in both cases. The larger value of \( n \) will be the required number of subintervals to achieve the desired accuracy.

Would you like me to calculate the values for \(T_4\), \(M_4\), and the error bounds step by step?

Find the approximations T4 and M4 for the following integral: ∫ from 1 to 2 of (31e^(1/x))/x dx. Estimate the errors in the approximations using error bounds for the trapezoidal and midpoint rules. How large should n be to achieve an accuracy of 0.0001?

Explore the numerical integration of ∫(31e^(1/x))/x dx using the trapezoidal and midpoint rules with four subintervals. Learn about error estimation and determining the appropriate number of subintervals for accuracy. Suitable for college-level calculus students.

Math Problem Statement

1 31e1⁄x dx T4

Correct: Your answer is correct. M4

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