Riemann Sums on the TI graphing calculators

Riemann Sums on the TI graphing calculators MATH 101FALL 2003

Riemann Sums on the TI graphing calculators

We can evaluate Riemann sums on the TI graphing calculators without doing any programming. We use the sum and seq functions to do this.
Recall that a Riemann sum is an expression of the form
where the x_i^* are sample points inside intervals of width .
In the examples below, we'll calculate with

f(x) = 1/(1+x²)
and = 0.01, so that the subintervals are

[0.00, 0.01], [0.01, 0.02], [0.02, 0.03], . . ., [0.99, 1.00]
We'll take the sample points to be the midpoints of the subintervals, so that

x₁^* = 0.005, x₂^* = 0.015, x₃^* = 0.025, . . ., x₁₀₀^* = 0.995,
Click on one of the links to see this worked out on a
[TI-83], [TI-85], or [TI-89].
On the TI-83
First we store the function 1/(1+x²) as Y₁ on the graph page. To do this, we press , clear whatever happens to be in Y₁, and enter 1/(1+X²).

Next we press to return to the Home screen.
We want to evaluate the sum of a sequence of values of Y₁, so we need the sum and seq functions. We press and then use the arrow keys to highlight MATH and press to select sum(.
To get the seq function, we again press and then use the arrow keys to highlight OPS. We press to select seq(.
Next we need to enter the function name Y₁. To do this, we press , select Y-VARS, and press twice.
At this point, the home screen shows sum(seq(Y₁. We want to multiply the value of Y₁ by , which in this case is 0.01. To do so, we just enter *0.01.
To tell the calculator what values of X to use, we complete the command to read

sum(seq(Y₁*0.01,X,0.005,0.995,0.01))
This says "Add up the values of Y₁*0.01 for X going from 0.005 to 0.995 in increments of 0.01." We press to get the result

On the TI-85
First we store the function 1/(1+x²) as y1 on the graph page. To do this, we press , then . We clear whatever happens to be in y1, and enter 1/(1+x²).

Next we press to return to the Home screen.
We want to evaluate the sum of a sequence of values of y1, so we need the sum and seq functions. We press (to select OPS) . The functions sum and seq appear on the menu at the bottom of the screen.
We press to get sum seq( in the display.
Next we enter the function name y1 by pressing .
We want to multiply the value of y1 by , which in this case is 0.01. To do so, we just enter *0.01.
To tell the calculator what values of x to use, we complete the command to read

sum seq(y1*0.01,x,0.005,0.995,0.01)
This says "Add up the values of y1*0.01 for x going from 0.005 to 0.995 in increments of 0.01." We press to get the result

On the TI-89
First we define a function f(x) as 1/(1+x²). To do this, we begin at the home screen and press and then to get the word Define in the entry line. We type in f(x)=1/(1+x^2) and then to define the function.

Now we need sum and seq. To get sum, we press (to select List) and (to select sum(). To get seq(, we press again, and then . The entry line on the home screen now says sum(seq(.
The sum we want is the sum of 0.01*f(x) as x goes from 0.005 to 0.995 in increments of 0.01. Here's how we complete the command line:

sum(seq(0.01*f(x),x,0.005,0.995,0.01))
We press and wait a few seconds for the calculator to produce an answer.

Notes
Under the F3 (Calc) menu, there's a summation command. This is more difficult to use than sum(seq(, and I don't recommend it.
If you give the limits on x as decimals, then the answer comes back as a decimal approximation. If you give limits as integers or fractions, then you need to press the green diamond to get an answer in decimal form.