HP 33S User manual

hp calculators
HP 33S Solving numeric integration problems
Numeric integration
Using the integration function
Practice solving numeric integration problems

hp calculators
HP 33S Solving numeric integration problems
Numeric integration
Numeric integration has many uses in different areas of science. One of the more common ways to visualize integration
is that of the area under a curve to the X-axis between two points.
Using the integration function
The HP 33S has a very powerful numeric integrator built into the calculator. This function is found above the #key
and is access by pressing º". The method used in this training aid will be to enter the function to integrate as an
equation and then to integrate it between an upper and lower limit of integration.
The general approach to integrate an equation will be:
Step 1: If the equation that defines the integrand's function isn't stored in the equation list, key it in and leave
Equation mode. The equation usually contains just an expression.
Step 2: Enter the limits of integration:
in RPN mode, key in the lower limit and press Ï, then key in the upper limit;
in algebraic mode, key in the lower limit, press w, then key in the upper limit.
Step 3: Display the equation: Press ºd and, if necessary, scroll through the equation list (press the Øor
×keys found on the cursor key at the top of the HP 33S) to display the desired equation.
Step 4: Select the variable of integration: Press º" and then press the appropriate key on the HP 33S to
indicate the proper variable (the 2key would choose X, for example). This starts the calculation.
Note that using the integration function uses much more of the calculator's memory than any other operation and,
although highly unlikely, if a MEMORY FULL message is shown, refer to appendix B in the HP 33S manual for more
information on what steps to take.
You can halt a running integration calculation by pressing Åor ¥. However, no information about the integration is
available until the calculation finishes normally.
The display format setting chosen through the Þmenu affects the level of accuracy assumed for your function
and used for the result. The integration is more precise but takes much longer in the ALL setting (Þ4) and in
the FIX (Þ1), SCI (Þ2), and ENG (Þ3) modes with more digits displayed. The uncertainty
of the result ends up in the Y–register, pushing the limits of integration up into the T– and Z–registers.
This training aid cannot begin toillustrate the wide range of applications available using the built-in numeric integration
function, but it can illustrate some of the more common uses. For additional information, see chapters 8 and 14 of the HP
33s RPN/ALG Scientific Calculator Owner’s Manual.
Practice solving numeric integration problems
Example 1: Integrate the function 1/X from 1 to 10. Use FIX 4 as the display setting.
Solution: In either RPN or algebraic mode: Þ14ºd,hXº|Ï
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hp calculators
HP 33S Solving numeric integration problems
The display should look similar to the one shown in Figure 1. Note, if you have other equations already in
the HP 33S calculator, the top line of the display may not indicate "EQN LIST TOP" but may show another
equation.
Figure 1
To show the checksum and length of this equation, press the following in RPN or algebraic mode. Note that
the symbol Õ means to press the right arrow direction of the cursor key at the top of the HP 33S.
In RPN or algebraic mode: ¹uÕϺÎ
Figure 2
If the checksum of the equation just entered does not equal B3AA, then you have not entered it correctly.
To exit equation mode, press:
In RPN or algebraic mode: ºd
Now enter the lower and upper limits of the integration.
In RPN mode: 1Ï10ºd
In algebraic mode: 1w10ºd
(Note: At this point, use the Øor ×keys to scroll through the equation list and make sure the
equation is displayed as shown in Figure 3 below). Note, if you have other equations already in the HP 33S
calculator, the top line of the display may not indicate "EQN LIST TOP" but may show another equation.
Figure 3
Integrate the function using X as the variable of integration.
º"X
After a few moments, the HP 33S will display the answer shown below.
Figure 4
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hp calculators
HP 33S Solving numeric integration problems
Now view the uncertainty of the result.
In RPN mode: Å
In algebraic mode: <
Figure 5
Answer: The area under the 1/X curve from 1 to 10 is approximately 2.3026. Figure 5 shows the uncertainty of the
result assuming algebraic mode. In RPN mode, the uncertainty is shown in the second level of the stack.
Example 2: Integrate the function Sin2(X) from 0 to ∏. Use FIX 4 as the display setting. Make sure the HP 33S is in
radians mode.
Solution: In either RPN or algebraic mode: Þ14Ý2
ºdkhXº|)2Ï
Note: It is possible to write the equation using the =function, but the equation displayed using the )
function is may be clearer to read. The display should look similar to the one shown in Figure 6.
Figure 6
To show the checksum and length of this equation, press the following in RPN or algebraic mode. Note that
the symbol Õ means to press the right arrow direction of the cursor key at the top of the HP 33S.
In RPN or algebraic mode: ¹uÕϺÎ
Figure 7
If the checksum of the equation just entered does not equal C615, then you have not entered it correctly.
To exit equation mode, press:
In RPN or algebraic mode: ºd
Now enter the lower and upper limits of the integration.
In RPN mode: 0Ϻjºd
In algebraic mode: 0wºjºd
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hp calculators
HP 33S Solving numeric integration problems
(Note: At this point, use the Øor ×keys to scroll through the equation list and make sure the
equation is displayed as shown in Figure 8 below). Note, if you have other equations already in the HP 33S
calculator, the top line of the display may not indicate "EQN LIST TOP" but may show another equation.
Figure 8
Integrate the function using X as the variable of integration.
º"X
After a few moments, the HP 33S will display the answer shown below.
Figure 9
Now view the uncertainty of the result.
In RPN mode: Å
In algebraic mode: <
Figure 10
Answer: The area under Sin2(X) from 0 to ∏is approximately 1.5708. The uncertainty of the result is 0.0002, as
shown in the Y level of the stack, in Figure 10 (assuming RPN mode).
Example 3: Integrate the function shown below from 0 to 2∏. Use FIX 4 as the display setting. Make sure the HP 33S
is in radians mode.
0.25COS(X)-1 1
+Figure 11
Solution: In either RPN or algebraic mode: Þ14Ý2
ºd1¯
ºy1ÃnhXº|Ù0Ë25º|
Ï
The display should look similar to the one shown in Figure 12.
Figure 12
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hp calculators
HP 33S Solving numeric integration problems
To show the checksum and length of this equation, press the following in RPN or algebraic mode. Note that
the symbol Õ means to press the right arrow direction of the cursor key at the top of the HP 33S.
In RPN or algebraic mode: ¹uÕϺÎ
Figure 13
If the checksum of the equation just entered does not equal BB03, then you have not entered it correctly.
To exit equation mode, press:
In RPN or algebraic mode: ºd
Now enter the lower and upper limits of the integration. Note that the algebraic keystrokes are to allow for
the computation of the upper limit of integration.
In RPN mode: 0Ϻj2¸ºd
In algebraic mode: 2¸ºjÏw0wºd
(Note: At this point, use the Øor ×keys to scroll through the equation list and make sure the
equation is displayed as shown in Figure 14 below). Note, if you have other equations already in the HP
33S calculator, the top line of the display may not indicate "EQN LIST TOP" but may show another
equation.
Figure 14
Integrate the function using X as the variable of integration.
º"X
After a few moments, the HP 33S will display the answer shown below.
Figure 15
Now view the uncertainty of the result.
In RPN mode: Å
In algebraic mode: <
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HP 33S Solving numeric integration problems
Figure 16
Answer: The area under the function from 0 to 2∏is approximately 8.3776. The uncertainty in the result is 0.0008.
Figure 16 assumes algebraic mode. In RPN mode, the uncertainty is shown in the second level of the
stack.
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