Problem 48
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 \(f(x)=x^{2}+1\) and the \(x\) -axis on the interval \([0,2]\).
Step-by-Step Solution
Verified Answer
Answer: The approximations appear to approach a limit if the calculated right Riemann sums for different values of n in the last column of the table are getting closer to a specific value.
1Step 1: Define the right Riemann sum formula
Right Riemann sum is calculated using the formula: $$R_n = \sum_{i=1}^{n}f(x_i)\Delta x$$ where \(R_n\) is the approximation of the area, \(n\) is the number of subintervals, \(x_i\) is the right endpoint of each subinterval, and \(\Delta x\) is the width of each subinterval.
2Step 2: Compute the width of each subinterval
The width of each subinterval \(\Delta x \) is the length of the interval divided by the number of subintervals: $$\Delta x = \frac{b-a}{n}$$, where \(a\) is the starting point and \(b\) is the end point of the interval. For our problem, \(a=0\), \(b=2\), and n varies as \(10\), \(30\), \(60\) and \(80\).
3Step 3: Calculate the right endpoints for each subinterval
For each value of \(n=10, 30, 60,\) and 80, we need to find the right endpoints \(x_i\) which are given by: $$x_i = a+i\Delta x$$
4Step 4: Calculate the function values for each right endpoint
For each \(x_i\), we have to calculate the function value \(f(x_i)\): $$f(x_i) = (x_i)^2 + 1$$
5Step 5: Calculate right Riemann sums for each value of n
Now, we will use the right Riemann sum formula mentioned in step 1 to calculate \(R_n\) for \(n = 10, 30, 60,\) and 80. We should get 4 approximated areas.
6Step 6: Organize the calculations in a table
We can organize all the calculations in a table, showing the approximations for \(n=10, 30, 60,\) and 80 subintervals.
| Number of Subintervals (n) | Width of Subintervals (\(\Delta x\)) | Right Endpoints (\(x_i\)) | Function Values (\(f(x_i)\)) | Right Riemann Sum (\(R_n\)) |
|------------------------|----------------------------|------------------|-----------------|----------------|
| 10 | 0.2 | ... | ... | ... |
| 30 | 0.0667 | ... | ... | ... |
| 60 | 0.0333 | ... | ... | ... |
| 80 | 0.025 | ... | ... | ... |
7Step 7: Comment on the approximations
Observe the calculated right Riemann sums for different values of n in the last column of the table. If the approximations are getting closer to a specific value, then the approximations appear to approach a limit.
Key Concepts
Area ApproximationInterval SubintervalsRight Endpoint Calculation
Area Approximation
When dealing with functions and their graphs, calculating the area under a curve is a common challenge. To simplify this process, we use the concept of area approximation. Imagine slicing the area between the curve and the x-axis into several rectangles. This idea helps in estimating the total area by adding up the areas of these rectangles.
The more rectangles (or subintervals) we use, the closer our approximation will be to the true area. This method is invaluable, especially when dealing with complex functions where finding an exact area is difficult or impossible.
The method of Riemann sums, specifically the right Riemann sum method, utilizes these rectangles where the height of each rectangle is determined by the value of the function at the right endpoint of the subinterval. As a result, by increasing the number of rectangles, our approximation becomes ever more precise.
The more rectangles (or subintervals) we use, the closer our approximation will be to the true area. This method is invaluable, especially when dealing with complex functions where finding an exact area is difficult or impossible.
The method of Riemann sums, specifically the right Riemann sum method, utilizes these rectangles where the height of each rectangle is determined by the value of the function at the right endpoint of the subinterval. As a result, by increasing the number of rectangles, our approximation becomes ever more precise.
Interval Subintervals
Interval subintervals are key to breaking down a given interval into smaller parts, which makes calculations more manageable and approximations more accurate. The interval we consider is [0, 2], meaning the function is evaluated between 0 and 2 on the x-axis. By dividing this main interval into "n" subintervals, we get a smaller individual section called a subinterval.
Each subinterval has a width (\(\Delta x\)), which is computed by dividing the length of the entire interval by the number of subintervals (\(n\)). The formula to find this width is \(\Delta x = \frac{b-a}{n}\), where \(a\) and \(b\) are the starting and ending points of the main interval, respectively.
The values of "n" chosen in our case (10, 30, 60, and 80) affect the width of these subintervals significantly, which directly influences the precision of the area approximation. More subintervals result in smaller widths and therefore more accurate estimations.
Each subinterval has a width (\(\Delta x\)), which is computed by dividing the length of the entire interval by the number of subintervals (\(n\)). The formula to find this width is \(\Delta x = \frac{b-a}{n}\), where \(a\) and \(b\) are the starting and ending points of the main interval, respectively.
The values of "n" chosen in our case (10, 30, 60, and 80) affect the width of these subintervals significantly, which directly influences the precision of the area approximation. More subintervals result in smaller widths and therefore more accurate estimations.
Right Endpoint Calculation
Using the right endpoint of each subinterval simplifies the approximation process. For every subinterval, the right endpoint is the point we choose on the x-axis to evaluate the height of our rectangle.
The formula for determining each right endpoint \(x_i\) is \(x_i = a + i\Delta x\), where \(a\) is the start of the interval, \(i\) is the subinterval index, and \(\Delta x\) is the width of each subinterval. This calculation is crucial as it determines where on the curve we assess the function's value to find the height of the rectangle.
Choosing the right endpoint helps provide a consistent method for area approximation in right Riemann sums. It is important to accurately compute these right endpoints for each subinterval, as this directly affects the summation of the area's approximation.
The formula for determining each right endpoint \(x_i\) is \(x_i = a + i\Delta x\), where \(a\) is the start of the interval, \(i\) is the subinterval index, and \(\Delta x\) is the width of each subinterval. This calculation is crucial as it determines where on the curve we assess the function's value to find the height of the rectangle.
Choosing the right endpoint helps provide a consistent method for area approximation in right Riemann sums. It is important to accurately compute these right endpoints for each subinterval, as this directly affects the summation of the area's approximation.
Other exercises in this chapter
Problem 48
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