f(x). These are quite simple to evaluate. All you do is use the right-hand or left-hand sum formula,
plugging in the appropriate values for f(x). One thing you should watch out for is that sometimes the x-
values are not evenly spaced, so make sure that you use the correct values for the widths of the rectangles.
Let’s do an example.
Example 4: Suppose we are given the following table of values for x and f(x).
Use a right-hand Riemann sum with 5 subintervals indicated by the data in the table to approximate
f(x) dx.
Recall that the formula for finding the area under the curve using the right endpoints is: [y 1 + y 2 +
y 3 +...+ yn]. Here, the width of each rectangle is 2. We find the height of each rectangle by evaluating f(x)
at the appropriate value of x, the right endpoint of each interval on the x-axis. Here, y 1 = 13, y 2 = 15, y 3 =
14, y 4 = 9, and y 5 = 3. Therefore, we can approximate the integral with f(x) dx = (2)(13) + (2)(15) +
(2)(14) + (2)(9) + (2)(3) = 108.
Let’s do another example, but this time the values of x will not be evenly spaced on the x-axis.
Example 5: Given the following table of values for x and f(x):
Use a left-hand Riemann sum with 6 subintervals indicated by the data in the table to approximate
f(x) dx.
Recall that the formula for finding the area under the curve using the left endpoints is [y 0 + y 1 + y 2
+...+ yn (^) − 1]. This formula assumes that the x-values are evenly spaced but they aren’t here, so we will