More about Integration | HP 33s Scientific specification

Because the calculation time depends on how soon a certain density of sample points is achieved in the region where the function is interesting, the calculation of the integral of any function will be prolonged if the interval of integration includes mostly regions where the function is not interesting. Fortunately, if you must calculate such an integral, you can modify the problem so that the calculation time is considerably reduced. Two such techniques are subdividing the interval of integration and transformation of variables. These methods enable you to change the function or the limits of integration so that the integrand is better behaved over the interval(s) of integration.

More about Integration

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HP 33s Scientific manual More about Integration

Contents

Hp 33s scientific calculator Printing History Contents Clearing the X-Register RPN The Automatic Memory Stack Storing Data into Variables How You Can Use Equations Summary of Equation Operations Fractions in Programs Accuracy of Integration Complex Stack Complex Operations ContentsSolving an Equation For More Information Programming 12-7 12-4Data Input and Output 12-4 Entering a Program 12-5 12-6 Branching GTO 13-4 13-113-2 13-3 15-12 Managing Calculator Memory ContentsPart 3. Appendixes and Reference Vector Operations 15-1 About ALG Resetting the Calculator Clearing MemoryRound-Off Error Contents Disabling Operations Neutral Operations Conditions That Prolong Calculation Time UnderflowHow the Integral Is Evaluated Page Basic Operation PartPage Turning the Calculator On and Off Adjusting Display ContrastGetting Started Important Preliminaries 2Getting Started Highlights of the Keyboard and DisplayShifted Keys Cursor Keys Alpha Keys 4Getting Started Silver Paint KeysBackspacing and Clearing ¾ when number entry is complete Keyboard-entry modeAre described in Using Menus on page 1-7. If Equation-entry mode 6Getting Started Numeric Functions Using MenusHP 33s Menus Chapter Name Programming Instructions To use a menu function Other functions8Getting Started Description KeysDisplay Exiting Menus RPN and ALG Keys RPN mode ALG modeTo select RPN mode To select ALG mode First Line Second Line Display and Annunciators RAD or HP 33s Annunciators Meaning Chapter12Getting Started 2 3 §,¨ Keys Display Description Keying in NumbersMaking Numbers Negative Exponents of Ten Other Exponent Functions Understanding Digit EntryKeying in Exponents of Ten 16Getting Started Doing ArithmeticRange of Numbers and Overflow Keys Display Key in the number. You dont need to press ‘ One-Number FunctionsTwo-Number Functions , \, , Qor T D, , \, ,Q Controlling the Display FormatPeriods and Commas in Numbers ‘3 z Scientific Format Number of Decimal PlacesFixed-Decimal Format % ALL Format SHOWing Full 12-Digit Precision20Getting Started Engineering Format KeysDisplayDescription FractionsEntering Fractions 22Getting Started DisplayDescription MessagesDisplaying Fractions 24Getting Started Calculator MemoryChecking Available Memory Clearing All of Memory RPN The Automatic Memory Stack RPN The Automatic Memory StackWhat the Stack Is 2RPN The Automatic Memory Stack X and Y-Registers are in the DisplayClearing the X-Register Rµ Roll Up Reviewing the Stack¶ Roll Down 4RPN The Automatic Memory Stack Arithmetic How the Stack Does ItExchanging the X- and Y-Registers in the Stack Lost How Enter Works Filling the stack with a constant Using a Number Twice in a RowHow Clear x Works 6RPN The Automatic Memory Stack Last X Register Wrong One-Number Function Correcting Mistakes with Last, \, , Qor T 8RPN The Automatic Memory Stack 96.704 + Reusing Numbers with Last› Mistake 10RPN The Automatic Memory Stack Work from the Parentheses Out Chain Calculations in RPN mode 12RPN The Automatic Memory Stack 16.3805x5 = 181.0000 SolutionExercises Order of Calculation ‘14 ‘ More Exercises14RPN The Automatic Memory Stack ⋅ 0.8 ⎟ 12.5 − 0.73 = Page Storing Data into Variables Storing Data into Variables Example Storing Numbers Storing and Recalling Numbers2Storing Data into Variables To recall a copy of a number from a variable to the display To review the values at any or all non-zero variables Viewing a Variable without Recalling ItReviewing Variables in the VAR Catalog 4Storing Data into Variables Clearing VariablesArithmetic with Stored Variables Storage Arithmetic Result 3 That is, x Recall Arithmetic 6Storing Data into Variables Exchanging x with Any VariableLzE Variable Page Real-Number Functions Real-Number FunctionsExponential and Logarithmic Functions 2Real-Number Functions Power FunctionsQuotient and Remainder of Division To Calculate Press To Calculate Press Result TrigonometryEntering π 4Real-Number Functions Setting the Angular ModeTrigonometric Functions Option Description Annunciator Programming Note NzR ‘x Q Hyperbolic FunctionsPercentage Functions 6Real-Number Functions 16.12 ‘ 15.76 T . 8Real-Number Functions Physics ConstantsConst Menu Items Description Value To insert a constant Conversion FunctionsItems Description Value To convert between rectangular and polar coordinates Coordinate Conversions10Real-Number Functions Example Conversion with Vectors Example Polar to Rectangular Conversion Example Converting Time Formats Time Conversions12Real-Number Functions To convert between decimal fractions and minutes-seconds To convert an angle between degrees and radians Angle ConversionsUnit Conversions To Convert Press Displayed Results Probability Probability FunctionsFactorial Gamma Display KeysExample Combinations of People 16Real-Number Functions Parts of Numbers Names of Functions Page Entering Fractions Fractions 2Fractions Fractions in the DisplayDisplay Rules Entered Value Internal Value Displayed Fraction Accuracy Indicators 4Fractions Changing the Fraction DisplayLonger Fractions Choosing a Fraction Format Setting the Maximum Denominator 6Fractions Examples of Fraction DisplaysTo Get This Fraction Format Change These Flags Fraction How 2.77 Is Displayed Format Denominator Fixed Rounding FractionsFraction Number Entered and Fraction Displayed Format ¼ 8Fractions Fractions in Equations Fractions in Programs Page Example Calculating with an Equation Entering and Evaluating EquationsHow You Can Use Equations Entering and Evaluating Equations 2Entering and Evaluating Equations ZNzZLL Key Operation Enters and leaves Equation modeSummary of Equation Operations To enter an equation Entering Equations into the Equation ListVariables in Equations 4Entering and Evaluating Equations Functions in Equations Numbers in Equations ZLC z Displaying and Selecting EquationsParentheses in Equations LR d To select an equation Editing and Clearing Equations——— To view a long equation 8Entering and Evaluating Equations Evaluating Equations Types of Equations To evaluate an equation Type of Equation Result for ‘10Entering and Evaluating Equations Fx ¼ Example Evaluating an Equation with Enter Using Enter for Evaluation Example Evaluating an Equation with XEQ Using XEQ for EvaluationResponding to Equation Prompts 12Entering and Evaluating Equations „ To display digits hidden by the prompt, press Syntax of EquationsOperator Precedence ‘5 g 14Entering and Evaluating Equations Order Operation ExampleEquations Meaning Σxy Equation Functions Perimeter = a + b + h Operation Equation function16Entering and Evaluating Equations Example Perimeter of a Trapezoid Area = HLA ’.25 zLN zLD 2 zR NqLN `qONqLN `‘Cos π / n Example Area of a Polygon Example Checksum and Length of an Equation Syntax ErrorsVerifying Equations 18Entering and Evaluating Equations Markup ⋅ Cost = Price Solving EquationsSolving an Equation Solving Equations Example Solving the Equation of Linear Motion LD dLVZLT › 2Solving Equations Value Example Solving the Ideal Gas Law Equation HL P zZL T 4Solving Equations Understanding and Controlling Solve LV q 6Solving Equations Verifying the Result Choosing Initial Guesses for Solve Interrupting a Solve Calculation 8Solving Equations Example Using Guesses to Find a Root ZL H 40 H 10Solving Equations For More Information Page Integrating Equations Integrating Equations 2Integrating Equations Integrating Equations ≥ FNJ0 x = π1 ≥0π cos x sin t dt Value ≥ 0π f t = ≥ t 4Integrating EquationsExample Sine Integral Accuracy of Integration Example Specifying Accuracy Specifying AccuracyInterpreting Accuracy 6Integrating Equations Example Changing the Accuracy Its uncertainty indicates 8Integrating Equations To enter a complex number Operations with Complex NumbersComplex Stack + iy To do an operation with one complex number Complex Operations2Operations with Complex Numbers To do an arithmetic operation with two complex numbers Functions for One Complex Number, z To Calculate Press 4Operations with Complex Numbers ‘2 ‘ ‘4 2 5 ‘ Using Complex Numbers in Polar Notation2 3 ‘ ‘2 Example Vector Addition 6Operations with Complex Numbers Operations with Complex Numbers Page Base Conversions and Arithmetic Base Conversions and ArithmeticBase Menu Menu label Description 10-2Base Conversions and Arithmetic Arithmetic in Bases 2, 8, Keys Display 10-4Base Conversions and Arithmetic Representation of NumbersNegative Numbers ˜˜ 7FFFFFFFF Range of Numbers 10-6Base Conversions and Arithmetic Windows for Long Binary Numbers Statistical Operations Statistical OperationsEntering Statistical Data 11-2Statistical Operations Correcting Errors in Data EntryEntering One-Variable Data Entering Two-Variable Data ‘20 Initial x, y Corrected x, y400 11-4Statistical Operations Statistical CalculationsMean Statistics Menus Key Description Number of Parts y Example Weighted Mean Two Variables· Price per Part 11-6Statistical Operations Sample Standard DeviationPopulation Standard Deviation Uº Linear Regression Menu Linear Regressionσº Keys DisplayDescription Grain Yield 11-8Statistical OperationsExample Curve Fitting Nitrogen Applied 70, y Limitations on Precision of Data Normalizing Close, Large Numbers Summation Values and the Statistics RegistersSummation Statistics 11-10Statistical Operations # ™ ´º¸/ ™ ´¸/ ™ ´º/ ™ ´¸/ ™ ´º/ ‡ Access to the Statistics RegistersStatistics Registers in Calculator Memory Example Viewing the Statistics Registers 11-12Statistical Operations Programming Page Example a Simple Program RPN mode 5 !Nz ALG mode 5 !zN‘Simple Programming Simple Programming 12-2Simple Programming RPN mode ALG modeRPN mode Display Description Program Labels Selecting a ModeDesigning a Program Program Boundaries LBL and RTN ALG Operations Using RPN, ALG and Equations in ProgramsData Input and Output Strengths of RPN Operations To enter a program into memory Entering a Program To enter an equation in a program line Keys That Clear12-6Simple Programming Example Entering a Labeled Program RPN modeFunction Names in Programs Example Entering a Program with an Equation HN ZLR 12-8Simple Programming NzX Testing a ProgramRunning a Program Executing a Program XEQ 12-10Simple Programming Example Testing a Program @ Using Input for Entering DataEntering and Displaying Data To use the Input function in a program 12-12Simple Programming To respond to a prompt Using View for Displaying Data Example INPUT, VIEW, and Messages in a Program Using Equations to Display Messages12-14Simple Programming = 2π R2 + 2π RH = 2π R R + H NzLR LR z › LH ›p 12-16Simple Programming Displaying Information without Stopping Interrupting a Running Program Error StopsStopping or Interrupting a Program Programming a Stop or Pause STOP, PSE To insert a program line Editing a Program12-18Simple Programming To delete a program line Viewing Program Memory Program Memory 12-20Simple Programming Memory UsageCatalog of Programs MEM Clearing One or More Programs To clear all programs from memory Checksum V Selecting a Base Mode in a ProgramNonprogrammable Functions Programming with Base Ax4 + Bx3 + Cx2 + Dx + E Decimal mode set Hexadecimal mode setPolynomial Expressions and Horners Method Numbers Entered in Program Lines 12-24Simple Programming Keys Display ALG mode‰X Checksum and length E41A Page Programming Techniques Programming TechniquesRoutines in Programs 13-2Programming Techniques Calling Subroutines XEQ, RTN Example a Nested Subroutine Nested Subroutines A2 + B2 Branching GTO13-4Programming Techniques Programmed GTO Instruction Using GTO from the Keyboard 13-6Programming Techniques Conditional Instructions Test Menus Tests of Comparison x?y, x?0 !- % ! ! ! º¸@ ! Program Lines Description RPN modeFlags 13-8Programming Techniques Set FlagFraction-Control Flags Status Clear 13-10Programming Techniques Annunciators for Set Flags Using FlagsFlags Menu Menu Key Description 13-12Programming Techniques Example Using Flags $ @ $ $ @ $ Program Lines Description RPN mode Example Controlling the Fraction Display 13-14Programming Techniques Program Lines Description 13-16Programming Techniques Loops . ! º6¸@ ! #$ ! Conditional Loops GTO ! ! Loop-Control Number Loops with Counters DSE, ISG13-18Programming Techniques $ $ $ ! $ 13-20Programming Techniques Indirectly Addressing Variables and LabelsVariable If i contains Then i will address Indirect Address ! L Program Control with13-22Programming Techniques Example Choosing Subroutines With Example Loop Control With If i holds Then XEQi callsProgram lines - L & %1L2 13-24Programming Techniques Equations with 1L2 L Page To solve a programmed function Solving and Integrating ProgramsSolving a Program Solving and Integrating Programs V= N x R x T Example Program Using ALG14-2Solving and Integrating Programs Pressure ⋅ volume. Pressure ⋅ volume 14-4Solving and Integrating Programs Example Program Using EquationLP z LV ’ LN z LR z LT ‘ @ @ ‘10 !@ # / Example Solve in a Program Using Solve in a Program14-6Solving and Integrating Programs / label #variable To integrate a programmed function Program Lines In RPN modeIntegrating a Program To write a program for ≥ FN 14-8Solving and Integrating Programs / label Using Integration in a ProgramSin Example ≥ FN in a Program 14-10Solving and Integrating Programs≥ G variable Restrictions on Solving and Integrating Page = R cosP = arctan Mathematics ProgramsVector Operations Mathematics Programs V1 = X i + Y j + Z k V2=U i + V j + W k 15-2Mathematics Programs= XU + YV + ZW Program Listing X2 + Y2 + Z2 and P 15-4Mathematics Programs Program Lines Description ALG mode Goes to the vector-addition routine 15-6Mathematics Programs Program Instructions Flags UsedRemarks Variables Used 15-8Mathematics Programs XR 15-10Mathematics Programs@ 07 g !@ 125 g @ AX + DY + GZ = J BX + EY + HZ = K CX + FY + IZ = L Solutions of Simultaneous Equations15-12Mathematics Programs % 15-14Mathematics Programs Mathematics Programs 15-16Mathematics Programs Flags Used Scratch 15-18Mathematics Programs @ @ @ Polynomial Root Finder15-20Mathematics Programs @ ⋅ the sign of JK a1/2 Where J = a3/2 = y0 /2X2 + J + Lx + K + M = X2 + J Lx + K M = 15-22Mathematics Programs Polynomial ! 15-24Mathematics Programs B1 = a3a1 4a0 Checksum and length C7A6B0 =a04a0 a32 a12 B2= -a2 JK. a1 15-26Mathematics Programs + L + M 15-28Mathematics Programs Order Terms and Coefficients 15-30Mathematics Programs %/ 15-32Mathematics Programs Coordinate Transformations Old coordinate System 15-34Mathematics Programs Checksum and length 8C82 Remark 15-36Mathematics Programs Ÿ @ 15-38Mathematics Programs Statistics Programs Statistics ProgramsCurve Fitting 16-2Statistics Programs = B + MIn Description 16-4Statistics Programs #$ #$ 16-6Statistics Programs 16-8Statistics Programs %@ 37.9 g &@ XS%@ 40.5 g &@ value 104.5 g %@ 38.6 g &@ / / %@ 16-10Statistics Programs/ Normal and Inverse-Normal Distributions Logarithmic Exponential Power 16-12Statistics Programs Function ª º ª -+. H % ! 16-14Statistics Programs XS@ Variable used temporarily to pass the value S ⋅ 2 π toInverse program 16-16Statistics Programs 2f − Grouped Standard Deviation 16-18Statistics Programs Updates ƒxi 2fi in register Increments or decrements N 16-20Statistics Programs XU/ %@ 13 g @ / Group/ %@ @ %@ 16-22Statistics Programs%@ %@ Miscellaneous Programs and Equations Miscellaneous Programs and EquationsTime Value of Money 17-2Miscellaneous Programs and Equations …LN `QLI ›LF z ›LI Solve instructions 17-4Miscellaneous Programs and Equations Keys Display Description RPN modešas Retains 5750 in B calculates monthly interest rate 17-6Miscellaneous Programs and Equations Prime Number Generator LBL Y 17-8Miscellaneous Programs and Equations #$ No test is made to ensure that the input is greater than 17-10Miscellaneous Programs and Equations Appendixes and Reference Page Answers to Common Questions Support, Batteries ServiceSupport, Batteries, and Service A-1 Calculator Support Changing the Batteries 2Support, Batteries, and ServiceEnvironmental Limits To install batteries Support, Batteries, and Service A-3 Fire. The batteries can burst or explode Testing Calculator Operation4Support, Batteries, and Service Do not mutilate, puncture, or dispose of batteries ™Ÿ˜—š Support, Batteries, and Service A-5Self-Test 6Support, Batteries, and Service Warranty Europe Country Telephone numbers ServiceSupport, Batteries, and Service A-7 America Country Telephone numbers 8Support, Batteries, and ServiceAsia Pacific Country Telephone numbers Canada Support, Batteries, and Service A-9Regulatory Information Japan 10Support, Batteries, and Service User Memory and the Stack User Memory and the StackManaging Calculator Memory 2User Memory and the Stack Resetting the Calculator Category DefaultClearing Memory 4User Memory and the Stack Status of Stack LiftDisabling Operations Neutral Operations EQN FDISP Vlabel nnnn € θ ,r Status of the Last X Register6User Memory and the Stack ALG Summary C-1 ALG SummaryAbout ALG 2ALG Summary Power FunctionsDoing Two-number Arithmetic in ALG Simple Arithmetic ALG Summary C-3 Percentage Calculations‘27 Q… 4ALG Summary Permutations and CombinationsQuotient and Remainder Of Division 16.12 T ALG Summary C-5 Keys Display Description 6ALG Summary Reviewing the Stack ALG Summary C-7 Coordinate Conversions 8ALG Summary Integrating an EquationQ3 z Complex Operations Operations with Complex NumbersALG Summary C-9 To view the result of complex operations 10ALG Summary 2 ›3 `` Q2 › ALG Summary C-11 ºy4 ÃË2Ë5 ¹cº  ºy3 ÃË  12ALG Summary Initial x, y Corrected x, y ALG Summary C-13 20 Page More about Solving D-1 More about SolvingHow Solve Finds a Root Function Whose Roots Can Be Found 2More about Solving Example An Equation With One Root Interpreting ResultsMore about Solving D-3 Example An Equation with Two Roots 4More about Solving -%. More about Solving D-5/ / Example Discontinuous Function LX `’1.5 6More about Solving More about Solving D-7 − 1= 8More about Solving When Solve Cannot Find a Root Example a Relative Minimum More about Solving D-9 Example An Asymptote 10More about Solving Example Find the root of the equation ⎟ x + 0.3 − 0.5 =#LX q LX ›3 ``… More about Solving D-11 Example a Local Flat Region 12More about Solving!1%ª1%-22 More about Solving D-13 Round-Off Error 14More about Solving Underflow More about Integration More about IntegrationHow the Integral Is Evaluated 2More about Integration Conditions That Could Cause Incorrect Results ≥0∞ xe −x dx 4More about Integration LX zLX ` ‡ More about Integration 6More about Integration Conditions That Prolong Calculation Time . 8More about Integration More about Integration Page Messages F-1 Messages Or program. This might take a while 2Messages # variable Messages F-3 Self-Test Messages 4Messages Operation Index G-1 Operation IndexName Keys and Description 2Operation Index ˜or —j h Polar to rectangular coordinates Operation Index G-3 4Operation Index P Operation Index G-5 M …Complex Subtraction 6Operation IndexGComplex Change sign ›Complex Operation Index G-7 DSE variable Mvariable 8Operation IndexRHyperbolic VRadians to degrees ?Fractional part Operation Index G-9Natural exponential Wlabel Nnnn 10Operation IndexGTO label Vlabel label Integer part ISG variable Lvariable ”I”Operation Index G-11 Input variable ‰variable Common logarithm 12Operation IndexLBL label •label Natural logarithm WDegrees to radians Operation Index G-13Permutations of n items FPause 14Operation Index Operation Index G-15 16Operation Index Operation Index G-17 18Operation Index 11-10 Given an x-value Operation Index G-19≠0? 13-7 If x=0, skips the next program Line ≤0? 13-7 20Operation Index Power Returns y raised to the xth power Index-1 Index Index-2 Index-3 Index-4 Index-5 Index-6 Index-7 Index-8 Index-9 Index-10 Index-11 RPN Index-12 Index-13 View Index-14 Index-15