Problem 25 Find the approximations \(L_{n},... [FREE SOLUTION]

Chapter 7: Problem 25

Find the approximations $L_{n}, R_{n}, T_{n},$ and $M_{n}$ for $n=5$ $10,$ and $20 .$ Then compute the corresponding errors $E_{L}, E_{R}, E_{T}$ and $E_{M} .$ (Round your answers to six decimal places. You may wish to use the sum command on a computer algebra system. What observations can you make? In particular, what happens to the errors when $n$ is doubled? $$ \int_{0}^{1} x e^{x} d x $$

Short Answer

Expert verified

Errors decrease as $ n $ doubles, indicating better accuracy.

Step by step solution

Set up the Integral

The integral to approximate is given as $ \int_{0}^{1} x e^{x} \, dx $. We are to find the left endpoint $ L_n $, right endpoint $ R_n $, trapezoidal $ T_n $, and midpoint $ M_n $ approximations for $ n = 5, 10, $ and $ 20 $.

Define the Function and Intervals

Define $ f(x) = x e^x $. For a given $ n $, the interval $[0, 1]$ is divided into $ n $ equal subintervals of width $ \Delta x = \frac{1}{n} $. So, for $ n = 5, 10, $ and $ 20 $, the widths are $ 0.2, 0.1, $ and $ 0.05 $, respectively.

Calculate Left Endpoint Approximation $ L_n $

For the left endpoint approximation, use the formula: \[ L_n = \Delta x \sum_{i=0}^{n-1} f(x_i) \] Where $ x_i = i \Delta x $. Compute $ L_n $ for $ n = 5, 10, $ and $ 20 $.

Calculate Right Endpoint Approximation $ R_n $

For the right endpoint approximation, use the formula: \[ R_n = \Delta x \sum_{i=1}^{n} f(x_i) \] Where $ x_i = i \Delta x $. Compute $ R_n $ for $ n = 5, 10, $ and $ 20 $.

Calculate Trapezoidal Approximation $ T_n $

For the trapezoidal approximation, use the formula: \[ T_n = \frac{\Delta x}{2} \left(f(x_0) + 2\sum_{i=1}^{n-1} f(x_i) + f(x_n) \right) \] Compute $ T_n $ for $ n = 5, 10, $ and $ 20 $.

Calculate Midpoint Approximation $ M_n $

For the midpoint approximation, use the formula: \[ M_n = \Delta x \sum_{i=0}^{n-1} f\left(x_i + \frac{\Delta x}{2}\right) \] Compute $ M_n $ for $ n = 5, 10, $ and $ 20 $.

Calculate Exact Integral Value

Calculate the exact value of the integral analytically. The integral $ \int_{0}^{1} x e^{x} \, dx $ can be found to be $ (1+e^{-1}) \approx 0.796201 $.

Compute Errors for Each Approximation

Compute the errors $ E_L = |L_n - I| $, $ E_R = |R_n - I| $, $ E_T = |T_n - I| $, and $ E_M = |M_n - I| $ for each $ n $, where $ I $ is the exact value. Round these errors to six decimal places.

Analyze the Error Trend

Observe the computed errors: as $ n $ doubles, the errors generally decrease, indicating improved accuracy with increased subintervals.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Left Endpoint Approximation

The Left Endpoint Approximation, often referred to as the Left Riemann Sum, is a technique used to estimate the value of definite integrals. In this method, we take the value of the function at the start of each subinterval.

To find the Left Endpoint Approximation, we first divide the interval \[0, 1\] into $n$ equal subintervals.
The width of each subinterval, $\Delta x$, is calculated as $\frac{1}{n}$.
Next, for each subinterval, the function value at the left endpoint is evaluated.
The values are summed and then multiplied by $\Delta x$ to give $L_n$, the left endpoint approximation.

This method provides a simple way to estimate area under a curve, but it may not always give the most accurate result. As seen with smaller $n$, a smaller number of subintervals can lead to larger errors. Thus, doubling $n$ reduces the error, making the approximation better.

Right Endpoint Approximation

The Right Endpoint Approximation, or Right Riemann Sum, is another method to approximate the value of integrals. This technique evaluates the function at the end of each subinterval rather than at the start.

Just like with the left endpoint, the interval \[0, 1\] is divided into $n$ equal parts.
The interval width remains $\Delta x = \frac{1}{n}$.
For each subinterval, calculate the function's value at the right endpoint.
Sum these values and multiply by $\Delta x$ to compute $R_n$, the right endpoint approximation.

Using the right endpoint can capture variations in the function differently than the left endpoint, sometimes giving a more accurate result. Similar to the left endpoint, increasing $n$ reduces the error. Both right and left methods may overestimate or underestimate depending on the function's nature.

Trapezoidal Approximation

The Trapezoidal Approximation offers a more nuanced approach by averaging the left and right endpoint values for each subinterval. This method uses trapezoids rather than rectangles.

The overall interval is still divided into $n$ smaller sections with $\Delta x = \frac{1}{n}$.
Both endpoints of each subinterval contribute to the height of the trapezoid.
Each trapezoid's area is calculated and all are summed up, adjusted by $\frac{\Delta x}{2}$.
This results in $T_n$, the trapezoidal approximation.

This method typically provides a balance between overestimating and underestimating errors seen in the more basic endpoint methods, offering improved accuracy. Error reduction with an increase in $n$ is also observed here.

Midpoint Approximation

In Midpoint Approximation, we take a slightly different approach by evaluating the function at the center of each subinterval. This often leads to very accurate results.

The interval \[0, 1\] is divided into $n$ equal subintervals, with width $\Delta x = \frac{1}{n}$.
Instead of using endpoints, we find the midpoint of each subinterval: $(x_i + \frac{\Delta x}{2})$.
The function is evaluated at each midpoint, summed, and multiplied by $\Delta x$ to calculate $M_n$.

Midpoint method tends to provide very accurate approximations. Like the other methods, as we double $n$, the error decreases, making this approach a strong choice for numerical integration tasks.

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Short Answer

Step by step solution

Set up the Integral

Define the Function and Intervals

Calculate Left Endpoint Approximation \( L_n \)

Calculate Right Endpoint Approximation \( R_n \)

Calculate Trapezoidal Approximation \( T_n \)

Calculate Midpoint Approximation \( M_n \)

Calculate Exact Integral Value

Compute Errors for Each Approximation

Analyze the Error Trend

Key Concepts

Left Endpoint Approximation

Right Endpoint Approximation

Trapezoidal Approximation

Midpoint Approximation

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