**Multiple Integration with MPP, the
Mathematics Plotting Program**

* Double Integration*:

__Step 1__. Start the MPP (2D version)

__Step 2__. Choose Item 9 in the list, then press
"Enter"

__Step 3__. Read the information on the blue screen, and
then press "Enter"

__Step 4__. Choose "Rectangular" or
"Polar" in Section F1 by typing 1 or 2

__Step 5__. Choose the order of integration in Section F2,
depending on the problem you are trying to solve

__Step 6__. Fill in the blanks; note the following:

-- use * for multiplication, and parentheses, if there is a
confusion in reading information; for instance, the expression
"*x*sin*y*" may return an error, so type *x**sin(*y*)

-- type "pi" for the corresponding greek letter;

-- when in polar coordinates, press "Ctrl-T" to type
the greek letter for *theta*

__Step 7__. When finished entering numbers and functions,
pressing "F9" should show a graph of the region of
integration shaded out, together with a numerical estimate of the
integral on the left of the screen. Press "H" for a
menu from which you may choose to do various things; in
particular, you may print your results.

* Triple Integration*:

Follow the same 7 steps as in double integration, with the following changes:

In Step 2, choose Item A;

In Step 4, choose between "Rectangular",
"Cylindrical" or "Spherical"

In Step 7, only the numerical estimate is shown. To graph the
region of integration, use either the MPP3D graphing program or
the DPGraph 2000 (the latter provides more options and
capabilities for 3D graphing).