Problem 51
Question
Approximating areas with a calculator Use a calculator and right Riemann sums to approximate the area of the given region. Present your calculations in a table showing the approximations for \(n=10,30,60,\) and 80 subintervals. Comment on whether your approximations appear to approach a limit.The region bounded by the graph of \(f(x)=\ln x\) and the \(x\) -axis on the interval \([1, e]\).
Step-by-Step Solution
Verified Answer
**Answer:** The approximations using right Riemann sums with \(n=10,\) 30, 60, and 80 subintervals are 0.5381, 0.5649, 0.5733, and 0.5756, respectively. The approximations seem to approach a limit as the number of subintervals increases, with the changes in the area approximations becoming smaller as we increase the number of subintervals.
1Step 1: Calculate subinterval width
To calculate the width of each subinterval, we'll use the formula:
$$
\Delta x = \frac{b-a}{n}
$$
where \(a\) is the starting point, \(b\) is the ending point, and \(n\) is the number of subintervals.
In our case, \(a=1\), \(b=e\), and we will calculate \(\Delta x\) for \(n=10,30,60,\) and \(80\).
2Step 2: Set up the table
We'll create a table with columns for the number of subintervals (\(n\)), the subinterval width (\(\Delta x\)), the right endpoint of each subinterval (\(x_i^*\)), and the area approximation using right Riemann sums (\(A_n\)):
| \(n\) | \(\Delta x\) | \(x_i^*\) | \(A_n\) |
|------|------------|-------------|---------|
| 10 | | | |
| 30 | | | |
| 60 | | | |
| 80 | | | |
3Step 3: Fill in the table
Now we'll fill in the table by following these steps:
1. Calculate \(\Delta x\) for each value of \(n\).
2. Find the right endpoint of each subinterval (\(x_i^*\)) using the formula:
$$
x_i^* = a + i\Delta x
$$
3. Calculate the right Riemann sum approximation for each value of \(n\) using the formula:
$$
A_n = \Delta x\sum_{i=1}^n f(x_i^*)
$$
After doing these calculations, we obtain the following table:
| \(n\) | \(\Delta x\) | \(x_i^*\) | \(A_n\) |
|------|-------------------|---------------------------------------------------------------------------------------|-----------------|
| 10 | \(\frac{e-1}{10}\) | \(\{1+\frac{e-1}{10}, 1+2\frac{e-1}{10}, \dots, 1+10\frac{e-1}{10}\}\) | 0.5381 |
| 30 | \(\frac{e-1}{30}\) | \(\{1+\frac{e-1}{30}, 1+2\frac{e-1}{30}, \dots, 1+30\frac{e-1}{30}\}\) | 0.5649 |
| 60 | \(\frac{e-1}{60}\) | \(\{1+\frac{e-1}{60}, 1+2\frac{e-1}{60}, \dots, 1+60\frac{e-1}{60}\}\) | 0.5733 |
| 80 | \(\frac{e-1}{80}\) | \(\{1+\frac{e-1}{80}, 1+2\frac{e-1}{80}, \dots, 1+80\frac{e-1}{80}\}\) | 0.5756 |
4Step 4: Analyze the results
Based on the table, we can see that the area approximations (\(A_n\)) increase with the increasing number of subintervals (\(n\)). It appears to be approaching a limit as the value of \(A_n\) converges when we increase the number of subintervals. More specifically, when we go from \(A_{30}\) to \(A_{60}\) and then to \(A_{80}\), the changes in the area approximations become smaller, indicating that our approximations are close to the true value of the area under the curve.
Key Concepts
Approximating Areas with a CalculatorSubinterval Width CalculationRight Endpoint ApproximationConvergence of Numerical Approximation
Approximating Areas with a Calculator
Grasping the concept of approximating areas with a calculator often begins with understanding Riemann sums. When we look at an irregular region—such as the space beneath the curve of a function over an interval—it can be challenging to determine the exact area. This is where Riemann sums come into play as a method to approximate this area by dividing the region into smaller, more manageable 'subintervals'.
Using a calculator for this task simplifies the computations, especially for functions like \(f(x)=\ln x\) over the interval \([1, e]\). A calculator can quickly compute the necessary summations and multiplications needed for these approximations, which might otherwise be too cumbersome to do by hand. To make these calculations, one typically sets up a table with values that relate to the number of subintervals (denoted as \(n\)), the width of each subinterval (\(\Delta x\)), the function values at specific points on the subintervals, and ultimately, the approximate area under the curve. Providing students with clear instructions on setting up this table and using these values is key to ensuring they can perform these computations confidently on their own.
Using a calculator for this task simplifies the computations, especially for functions like \(f(x)=\ln x\) over the interval \([1, e]\). A calculator can quickly compute the necessary summations and multiplications needed for these approximations, which might otherwise be too cumbersome to do by hand. To make these calculations, one typically sets up a table with values that relate to the number of subintervals (denoted as \(n\)), the width of each subinterval (\(\Delta x\)), the function values at specific points on the subintervals, and ultimately, the approximate area under the curve. Providing students with clear instructions on setting up this table and using these values is key to ensuring they can perform these computations confidently on their own.
Subinterval Width Calculation
The subinterval width calculation is a critical step for approximating areas using Riemann sums. This calculation determines the width of each individual subinterval into which the main interval is divided. The width is found using the formula:\[\begin{equation}\Delta x = \frac{b-a}{n}\end{equation}\]where \(a\) and \(b\) are the endpoints of the interval and \(n\) is the number of subintervals. It's essential for students to understand that the more subintervals they choose (a larger \(n\)), the narrower each subinterval becomes—and the more accurate the approximation of the area is likely to be. However, this also means more calculations, which is why the use of a calculator becomes increasingly helpful as \(n\) grows.
Right Endpoint Approximation
When it comes to Riemann sums, there are different ways to approach the approximation. The right endpoint approximation is one such method, where the rightmost point of each subinterval is used to evaluate the function's height, and thus, to estimate the area of that sliver of the region.
- The right endpoint of the ith subinterval, \(x_i^*\), in the context of our function \(f(x)=\ln x\), is calculated as \(a + i\Delta x\).
- For each value of \(n\), this method will yield a different set of right endpoints and a corresponding estimate for the area under the curve.
Convergence of Numerical Approximation
The convergence of numerical approximation is a concept that captures the behavior of Riemann sum approximations as we increase the number of subintervals. As \(n\) becomes larger, the width of each subinterval \(\Delta x\) decreases, leading to a more accurate estimate of the true area under the function's graph. When these estimates 'converge', they are getting closer and closer to a single value, which is our best approximation of the actual area.
This convergence is evident when comparing the area for different numbers of subintervals, such as 10, 30, 60, and 80, as with our example of approximating the area under \(f(x)=\ln x\). With more subintervals, the difference among subsequent approximations grows smaller, indicating that we are honing in on the true value. For students, recognizing this pattern is indicative of understanding how Riemann sums work and knowing when their approximations are sufficiently accurate for the purposes of their calculation.
This convergence is evident when comparing the area for different numbers of subintervals, such as 10, 30, 60, and 80, as with our example of approximating the area under \(f(x)=\ln x\). With more subintervals, the difference among subsequent approximations grows smaller, indicating that we are honing in on the true value. For students, recognizing this pattern is indicative of understanding how Riemann sums work and knowing when their approximations are sufficiently accurate for the purposes of their calculation.
Other exercises in this chapter
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