Approximating Riemann Sums with a Table: Step-by-Step Guide to Calculating Areas Under Curves

When given a table to approximate Riemann Sums…

When given a table to approximate Riemann Sums, you can use the table values to estimate the area under a curve by dividing it into smaller rectangles

When given a table to approximate Riemann Sums, you can use the table values to estimate the area under a curve by dividing it into smaller rectangles. This method is known as the rectangular approximation method.

To approximate the Riemann Sums using a table, follow these steps:

Step 1: Gather the necessary information
First, ensure that you have a table that contains the x-values and their corresponding y-values for the function you want to approximate. For example, consider the following table:

| x | y |
|——|——|
| 1 | 3 |
| 2 | 4 |
| 3 | 5 |
| 4 | 7 |
| 5 | 9 |

Step 2: Determine the width of the intervals
Find the difference between consecutive x-values in the table. This difference represents the width of each interval. In our example, the interval width is 1 since the difference between adjacent x-values is 1.

Step 3: Choose the type of Riemann Sum to approximate
There are three types of Riemann Sums: left-hand sum, right-hand sum, and midpoint sum. Each of these approaches uses different points within each interval to determine the height of the rectangles. Choose the type that suits the problem you are solving.

Step 4: Calculate the height of the rectangles
Depending on the chosen approach (left-hand, right-hand, or midpoint sum), use the corresponding y-values from the table to calculate the height of each rectangle. For instance, let’s use the left-hand sum approach:

For the first interval [1, 2], the y-value to use would be 3 (the leftmost y-value).

For the second interval [2, 3], the y-value to use would be 4 (the leftmost y-value).

And so on for the remaining intervals.

Step 5: Calculate the area of each rectangle
Multiply the width of each interval by the corresponding height from the previous step. This gives you the area of each rectangle.

Step 6: Find the sum of all the rectangle areas
Add up the areas of all the rectangles from the previous step to obtain an approximate value of the total area under the curve.

In this way, you can use the given table to approximate the Riemann Sums by dividing the area under the curve into smaller rectangles and calculating their areas. The more rectangles you use, the closer your approximation will be to the actual area.

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