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HP 50g Numeric Integration
Methods used
Integration commands
Substitution commands
Expansion commands
Numeric evaluation commands
Practice solving nume ric integration prob lems
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HP 50g Calculator: Numeric Integration Methods and Commands, Notas de estudo de Mecatrônica
An overview of the hp 50g calculator's capabilities for numeric integration, including methods used, integration commands, substitution commands, and expansion commands. It also includes examples of solving numeric integration problems using the hp 50g.
Tipologia: Notas de estudo
Antes de 2010
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HP 50g Numeric Integration Methods used Integration commands Substitution commands Expansion commands Numeric evaluation commands Practice solving numeric integration problems
HP 50g Numeric Integration Methods used The HP 50g provides several integration commands that we can use symbolic or numeric integration. This training aid considers two methods of numeric integration. The first is to directly evaluate numerically a given integral, while the second is to find a symbolic antiderivative, then substitute values for the variables, and evaluate it to a number. Integration commands The provided integration commands are INT, INTVX, RISCH and. Any of these commands can be used for symbolic integration in combination with substitution, expansion, and so on. The command INT is accessible using the built-in command catalogue of the HP50g. Press @Nto open the catalogue. From the catalogue you can select and execute any of the existing commands. The catalogue is much like a menu of an application, where you can use the arrow keys to select menu items, or jump to the items typing the first few letters of them. While the catalogue is active, press ~~int to jump to the command INT. Pressing the key or the menu key %OK% will put the selected item on the command line (or execute the selected item if RPN mode is on). Pressing CANCL will quit the command catalogue without executing the selected item. The command INT needs three arguments: The expression to be integrated, the variable of integration, and the value of the variable of integration where the antiderivative will be evaluated. The commands INTVX and RISCH are available in the menu "Derivatives and Integrals" This menu is accessed pressing !Öto open the "Calculus" menu. Figure 1 The first menu item is 1.DERIV & INTEG.... and it is highlighted (selected). In this CHOOSE box selecting 1.DERIV & INTEG... and pressing or %OK% takes you to a new menu which contains differentiation and integration commands: Figure 2 The commands INTVX and RISCH are in the second page of the menu, so you must press 7 to have the CHOOSE box scroll down and see them. The command INTVX is provided as a shorter way to perform integrations as it only needs one argument, the expression to be integrated, and uses automatically the current CAS variable VX (usually X) as the variable of integration. RISCH needs two arguments: the expression to be integrated and the variable of integration. Finally, the command is accessible from the keyboard pressing @Á. It needs four arguments: the lower and upper limit of integration, the expression that must be integrated, and the variable of integration. In many cases, this will be the command that is the best choice for numeric integration. The substitution commands The commands for substitution are SUBST, | (where) , and PREVAL. The command PREVAL allows for the substitution and evaluation of the difference g(x 2 )-g(x 1 ), where g(x) is the antiderivative of some function f(x) that we want to integrate between the limits x 1 and x 2. This command resides in the menu 1.DERIV & INTEG..... The command SUBST allows for the substitution of the variable of integration, since it will take care of altered integration limits and other necessary substitutions in the integral. This command resides in (the second page) of the menu "Algebra" which you access by pressing @×.
HP 50g Numeric Integration Figure 5 Answer: 1. Example 2: Find the numeric value of the integral: Solution: Assume RPN mode, CHOOSE boxes, variable X as the current VX and standard display format of numbers. Enter the integral using the command catalogue. Press @O@N~~in to jump to the first command that starts with "INT". Now the command INCR should be selected. The fourth command after INCR is the command INT. Press ˜˜˜˜ to select it. Figure 6 Press to put the command INT in the EquationWriter. Now continue the formula. !¸W~yQ2™~y™X Figure 7 We use Y instead of X in the function and in the variable of integration, and we keep X as the value that will be used to supply the limits of integration. Actually any other variable name could be used in the function and in the variable of integration. If your current VX is not X but, say A, you just supply that variable name as the third argument of INT. Press to put the expression on stack level 1. Enter the limits of integration. 0!ì
HP 50g Numeric Integration Figure 8 Evaluate the integral at the limits of integration and build-up the difference. !Ö1 The command PREVAL evaluates an expression of the current variable VX at two particular values and then calculates the difference between the two evaluated expressions. It takes three arguments: The expression on stack level 3, the lower value at stack level 2 and the higher value at stack level 1. The fact that PREVAL substitutes values only for the current VX was the reason for using variable Y instead of X and leaving variable X (usually the current VX) as a variable that PREVAL can use. The result of this operation is an unevaluated integral, which can be evaluated numerically. The HP 50g displays now: Figure 9 @ï (Calculate the numeric value) Figure 10 Answer:. Note: Even if the HP50g can integrate symbolically some function, the expression INT(function, vvariable, value) is only determined up to an additive constant. The anti derivative of a function is not a single function but a family of functions that differ from each other by some constant. It is the difference INT (function, variable, value 2 ) – INT (function, variable, value 1 ) that is unambiguous. Example 3: Find the numeric value of: Solution: Assume algebraic mode. Enter the integral.
HP 50g Numeric Integration Solution: Assume RPN mode, CHOOSE boxes, standard numeric formating, and X as the current variable VX. Put the expression on stack level 1. @O!Ö8XSX™™-TFigure 15 …×2 (Solve the integral symbolically.) Figure 16 Calculate the difference of the anti derivative at the limits of integration. 0!ì!Ö1 Figure 17 The answer is . Use @ï to get a numeric approximation of . Answer: 3. Note: Any of the four integration commands (not only INTVX) can be used in such cases where symbolic solutions of an integral can be found by the HP50g.