# Double integrals with a special Ti-84 Basic program

## Explanation of the program DOUBINT.8xp

Your Ti-84 plus can only integrate a function f(x) to x once. In physics, it is possible that you have to integrate twice to get the desired result. For example, the formulas for linear and rotating systems show that you have to integrate twice to get position out of acceleration.

First, we demonstrate the program by integrating cos(2πx) twice.The program uses the trapezoidal rule. For the step size dx the rules yield as for the differential equation solvers. For the step size dx: Xmax/300 usually satisfy. More info about step size, go to :

The program FUNCGEN can be used to generate a complex input function f(x). For info about the program FUNCGEN:

## Example : relationship between acceleration, velocity and position

## The program DOUBINT.8xp with input from the Function generator

Example: In positioning systems as a servo system, the acceleration profile shown in the figure below may be used. For one third of the time, there is an acceleration. For one third the acceleration is zero and one third of the time there is deceleration.

With the program Funcgen, this profile can be easily created. With this profile, it can be demonstrated that copper losses in an actuator are minimized for a desired displacement.

In this example, positioning time tmax is set to 3 sec. and acceleration is 1 m/sec^2 and deceleration is -1 m/sec^2.

The program calculates velocity and displacement as function of time(x=t). The results of velocity (Trapezoid) and displacement (S-curve) are displayed.In window mode on the TI-84, one can adjust the scaling, and in Y mode, one can select which plots to display. Values can be read out using the Trace command. First, the input made with the FUNCGEN program is displayed.

Download the program