The left sum of a given function f over the partition defined above is the Riemann sum where, for each j, the number tj is the left endpoint of the interval that runs from to In other words, we define the left sum by pointing at t he equation and clicking on Define and New Definition. Similarly, the right sum of f is and we define it by pointing at the equation and clicking on Define and New Definition. The arithmetic mean of the left and right sums is the trapezoidal sum which we define by pointing at the equation
Alternatively we could observe that
and use this equation for the definition of the trapezoidal sum. As we shall see from the examples that follow, the trapezoidal sum is frequently a much better approximation to the integral than either the left or the right sum. An even better approximation than the trapezoidal sum is the midpoint sum which we define by pointing at the equation
In this sum the function f is evaluated for each j at the midpoint of the interval that runs from to
Finally, the Simpson sum of f over the given partition is defined by pointing at the equation
As you may know, the Simpson sum is used only when the number n is even.