In a Midpoint Riemann Sum, we approximate the area using rectangles. The width or base of each rectangle is equal to the value of delta x (Δx=(b - a)/n). The height of each rectangle is equal to the value of the function at the midpoint between the two endpoints of its base. After evaluating the area for each rectangle, we add them all together to get an approximation of the area under the curve of a function.
Step 1:
Find the value of n, the number of rectangles being used. When given a table of values, n is equal to the number of coordinates given in the table, starting from the second x value given. When given a graph, n is generally given.
Step 2:
Find the value of delta x (Δx), the width of each rectangle. This is equal to the following equation.
Δx = (b - a)/n
• The variable b represent the last x-value of the interval given
• The variable a represent the initial x-value of the interval given
• (b-a) represents the space between the interval
• The variable n represents the number of rectangles within the space between the interval
When the coordinates given have varying distances from each other, Δx/width has a different value for each rectangle. This means that we must use the equation above to calculate the width for each individual rectangle.
Step 3:
Gather the value of each y-value for each midpoint between its base. These y-values will be used to get the height of the rectangle used to find the area. Each rectangle will have an area of Δx • f(x), each rectangle having their own value of Δx if needed.
Step 4:
After getting the area of each rectangle, we must add each of the areas together to get an approximation of the area under the curve of a function.