In a Trapezoidal Riemann Sum, we approximate the area using trapezoids. The width or base of each trapezoid is equal to the value of the sum of the two endpoints. The height of each trapezoid is equal to the value of delta x. After evaluating the area for each trapezoid, 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 trapezoid 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 height of each trapezoid. 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 trapezoid within the space between the interval
When the coordinates given have varying distances from each other, Δx/height has a different value for each trapezoid. This means that we must use the equation above to calculate the height for each individual trapezoid
Step 3:
Gather the value of the sum of each endpoint’s y-value. These sums will be used to get the width of the trapezoid used to find the area. Each trapezoid will have an area of (½)Δx • ( f(b) - f(a) ), each trapezoid having its own value of Δx if needed.
Step 4:
After getting the area of each trapezoid, we must add each of the areas together to get an approximation of the area under the curve of a function.