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Approximating Definite Integrals on a Graphing Calculator

Instructor: Michael Eckert

Michael has a Bachelor's in Environmental Chemistry and Integrative Science. He has extensive experience in working with college academic support services as an instructor of mathematics, physics, chemistry and biology.

Many definite integrals can be either too tedious or complex to solve for by hand. As a result graphing calculator technology has bridged the gap. With the use of the Texas Instruments (TI-83/84), such finite integrals can be approximated to a small fraction of a decimal.

Using a Graphing Calculator to Approximate Definite Integrals

With the use of a graphing calculator, one can approximate definite integrals to a very small fraction of a decimal. Two such methods implementing the TI-84 exist in this case:

  1. where a simple finite integral command may be used to approximate this value
  2. or where a graphing feature coupled with an integration command might be used.

Before using these two methods, let's make sure the functioning MODE of our calculator is set up correctly for our purposes:


mode

The calculator MODE is set correctly regarding setting e.g. whether we can deal with decimals or just whole numbers, radians or degrees, general functions or parametric equations, real numbers or imaginary, etc. Above is the displayed image for the MODE settings we desire for this exercise highlighted in bold.

Using the Finite Integration Command to Approximate a Definite Integral

Given the equation of the function to be integrated and the lower and upper limits through which to integrate this function, the finite integral command might be used to solve for a definite integral. Let's take y = cosx for instance. Let's calculate the value of the integral of y = cosx from x = -1.05 to x = 1.05:

To find a value for this integrand, we select the MATH command. After scrolling down to the finite integral command (highlighted below), we press ENTER:


mat

fnInt will be displayed on the home screen and we must fill in the following:


sc

Where cosx is the integrand function, x is the variable of integration and x = -1.05 to x = 1.05 are our lower to upper limits. After filling in these arguments, we press ENTER. We find that area under the curve y = cosx from x = -1.05 to x = 1.05 is approximately 1.73.


sc2

Note that the places to the right of the decimal point are nine, as we left MODE on float -as seen above; therefore, we can achieve a near exact value for a definite integral by these means.

Graphing Feature Coupled with the Integration Command to Approximate an Integral (definite)

Before using the graphing feature, it is beneficial to set the Window. Below is a standard which seems to work for many types of (but not all) functions with varying domains and ranges:


w

We set up WINDOW as follows: Xmin = -10, Xmax = 10, Xscl = 1, Ymin = -10, Ymax = 10, Yscl = 1 and Xres = 1. Xscl and Yscl determine how many tic-marks that we will have along the x and y-axis. Along each axis, we will have 10 in the negative direction from the origin (0,0) and 10 in the positive direction from the origin along both x and y-axis. Xres is a variable we can just take to be 1.

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