6More about Integration | HP 35s Scientific specification

Note that the rapidity of variation in the function (or its low–order derivatives) must be determined with respect to the width of the interval of integration. With a given number of sample points, a function f(x) that has three fluctuations can be better characterized by its samples when these variations are spread out over most of the interval of integration than if they are confined to only a small fraction of the interval. (These two situations are shown in the following two illustrations.) Considering the variations or fluctuation as a type of oscillation in the function, the criterion of interest is the ratio of the period of the oscillations to the width of the interval of integration: the larger this ratio, the more quickly the calculation will finish, and the more reliable will be the resulting approximation.

f (x)

Calculated integral of this function will be accurate.

x

ab

f (x)

Calculated integral of this function may be inaccurate.

x

ab

E-6More about Integration

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HP 35s Scientific manual 6More about Integration

Contents

HP 35s scientific calculator Users guide Printing History Contents Viewing a Variable Contents RPN The Automatic Memory StackStoring Data into Variables Trigonometry Entering π Real-Number FunctionsQuotient and Remainder of Division Hyperbolic Functions Fractions Entering and Evaluating Equations Operations with Complex Numbers Solving EquationsIntegrating Equations Vector Arithmetic 10-1 10-7 Base Conversions and Arithmetic and Logic 11-1Statistical Operations 12-1 Creating Vectors from Variables or Registers 10-8 Part 2. Programming Simple Programming 13-1 Programming Techniques 14-1 Solving and Integrating Programs 15-1 Support, Batteries, and ServicePart 3. Appendixes and Reference Statistics Programs 16-1 User Memory and the Stack ALG SummaryMore about Solving Messages Operation Index Round-Off ErrorMore about Integration How the Integral Is Evaluated Contents Part Page Important Preliminaries Adjusting Display ContrastGetting Started Turning the Calculator On and Off Highlights of the Keyboard and Display Shifted Keys2Getting Started Function Key Alpha KeysCursor Keys Left-shifted function Right-shifted Letter for alphabetic Keys for Clearing  Clear or CancelBackspacing and Clearing 4Getting Started Key Description Clear menu       HP 35s Menus Chapter Name Using Menusˆ ˆ    6Getting Started Programming Instructions Other functions  Exiting Menusor  8 Keys Display RPN and ALG Modes KeysDisplay RPN mode ALG mode 10Getting Started Undo key Annunciators Display and AnnunciatorsFirst Line Second Line 12Getting Started HP 35s Annunciators Meaning Chapter EQN 14Getting Started Keying in Numbers Making Numbers NegativeExponents of Ten  z16Getting Started  Understanding Entry CursorRange of Numbers and Overflow  Performing Arithmetic Calculations Single Argument or Unary Operations18Getting Started Two Argument or Binary Operations Key RPN,RPN Program ALG, Equation, ALG Program   Õ Controlling the Display Format Key RPN, RPN Program ALG, Equation, ALG Program 22Getting Started Periods and Commas in Numbers   8 Complex number display format  ,  , 8 24Getting Started 8× SHOWing Full 12-Digit Precision8    Fractions Entering Fractions26Getting Started  É Messages  Calculator Memory Checking Available Memory28Getting Started Clearing All of Memory 30Getting Started RPN The Automatic Memory Stack What the Stack IsRPN The Automatic Memory Stack 2RPN The Automatic Memory Stack X and Y-Registers are in the Display Clearing the X-RegisterReviewing the Stack Exchanging the X- and Y-Registers in the Stack Press 4RPN The Automatic Memory Stack Arithmetic How the Stack Does It How Enter Works 6RPN The Automatic Memory Stack How to Clear the Stack Filling the stack with a constant Last X Register 8RPN The Automatic Memory Stack Correcting Mistakes with Last  Reusing Numbers with Last Ù     Chain Calculations in RPN ModeWork from the Parentheses Out      ExercisesOrder of Calculation       More Exercises       18RPN The Automatic Memory Stack A Keys DisplayDescriptionStoring Data into Variables Press letter-key  Storing and Recalling Numbers2Storing Data into Variables Press letter-key  A G   VAR catalog Using the MEM CatalogViewing a Variable 4Storing Data into Variables C D E 6Storing Data into Variables Arithmetic with Stored VariablesStorage Arithmetic Storage arithmetic uses , , , or  Recall Arithmetic D Exchanging x with Any Variable D E Variables I and J A 10Storing Data into Variables Exponential and Logarithmic Functions Real-Number Functions   Power FunctionsQuotient and Remainder of Division   Trigonometry Entering π Option Description Annunciator Setting the Angular ModeTrigonometric Functions 4Real-Number Functions   Hyperbolic Functions Percentage Functions     Physics Constants Const Menu Items Description Value8Real-Number Functions Items Description Value Conversion Functions Rectangular/Polar Conversions10Real-Number Functions 30o 8 612Real-Number Functions Time Conversions Angle Conversions Unit Conversions Gamma Probability FunctionsFactorial Probability  16Real-Number Functions Fractional part Parts of NumbersInteger part Absolute value To calculate PressDisplay18Real-Number Functions Fractions Entering Fractions  Fractions in the Display Display Rules2Fractions Accuracy Indicators Entered Value Internal Value Displayed Fraction Setting the Maximum Denominator Changing the Fraction Display4Fractions  Choosing a Fraction Format 6Fractions   j To Get This Fraction Format Change These Flags Rounding Fractions Examples of Fraction Displays Fractions in Equations Fractions in Programs 10Fractions Entering and Evaluating Equations Entering and Evaluating EquationsHow You Can Use Equations = .25 π d2 l    D  2Entering and Evaluating EquationsValue   Summary of Equation Operations Key Operation Entering Equations into the Equation List Variables in Equations4Entering and Evaluating Equations Numbers in Equations Functions in Equations   Displaying and Selecting EquationsParentheses in Equations  Ö  Editing and Clearing Equations 8Entering and Evaluating Equations Types of Equations Evaluating Equations 10Entering and Evaluating Equations Using Enter for Evaluation Type of Equation Result for  Result for  Using XEQ for Evaluation      12Entering and Evaluating Equations Responding to Equation Prompts  Syntax of Equations Operator Precedence14Entering and Evaluating Equations Order Operation Example Equations Meaning Equation Functions 16Entering and Evaluating EquationsΣxy RPN Operation Equation function Perimeter = a + b + h sin1θ + sin1φ 18Entering and Evaluating Equations Area = Syntax Errors Verifying Equations Õ 20Entering and Evaluating Equations Solving Equations Solving EquationsSolving an Equation Markup ⋅ Cost = Price 2Solving Equations = v0 t + 1/2 g t G    P       P    Solving built-in Equation 6Solving Equations Understanding and Controlling Solve Verifying the Result Interrupting a Solve Calculation Choosing Initial Guesses for Solve8Solving Equations Solving Equations Type in the equation 10Solving Equations= 40 H ⋅ 20 H ⋅ 4 ⋅ H Equation 4 H  H  value For More Information 12Solving Equations Integrating Equations Integrating Equations Integrating Equations FN 2Integrating Equations J0 x = π 0 cosx sin t dt    = t4Integrating Equations Sinx X X Interpreting Accuracy Accuracy of IntegrationSpecifying Accuracy 6Integrating Equations 8        8 8Integrating Equations Operations with Complex Numbers Form  Complex Stack Complex Operations2Operations with Complex Numbers Functions for One Complex Number, z To Calculate Press Display Description 8  68  6 6 6 4Operations with Complex Numbers Using Complex Numbers in Polar Notation ?  ?  ?  6Operations with Complex Numbers Complex Numbers in Equations 6 ? ?    Program lines ALG modeComplex Number in Program        Vector Arithmetic Vector operationsVector Arithmetic  Õ 3    Absolute value of the vector  3  3 3 Dot product   10-4Vector Arithmetic   Angle between vectors Õ 3   Vectors in Equations 3  10-6Vector Arithmetic Vectors in Programs Program lines Description Creating Vectors from Variables or Registers Then hC Õ hA10-8Vector Arithmetic Menu label Description Base Conversions and Arithmetic and LogicBase Menu Base Conversions and Arithmetic and Logic    11-2Base Conversions and Arithmetic and Logic Õ  Arithmetic in Bases 2, 8, Logic Menu11-4Base Conversions and Arithmetic and Logic            Representation of NumbersNegative Numbers  Range of Numbers  Using base in program and equationsWindows for Long Binary Numbers    Statistical Operations Entering Statistical DataStatistical Operations Entering Two-Variable Data Correcting Errors in Data EntryEntering One-Variable Data 12-2Statistical Operations   Initial x, y Corrected x, y Statistics Menus Statistical CalculationsMean Menu Key Description      Sample Standard Deviation     Population Standard Deviation Linear Regression  ˆ Õ ˆ Linear Regression Menu12-8Statistical Operations        Limitations on Precision of Data 12-10Statistical Operations Summation Values and the Statistics Registers Summation Statistics Access to the Statistics Registers 12-12Statistical Operations Statistical Operations 12-14Statistical Operations Programming Page Simple Programming Simple Programming  13-2Simple ProgrammingArea = πx2 Designing a Program Selecting a ModeKeysDisplay In ALG mode  X  13-4Simple Programming Using RPN, ALG and Equations in ProgramsProgram Boundaries LBL and RTN  letter-key Data Input and Output Strengths of RPN Operations Entering a Program 13-6Simple Programming Clear functions and backspace key Function Names in Programs A13-8Simple Programming R RPN modeE  R   Running a ProgramExecuting a Program XEQ 13-10Simple Programming Testing a Program Keys Display Description RPN mode Entering and Displaying Data 13-12Simple Programming Using Input for Entering Data RPN mode ALG mode 13-14Simple Programming Using View for Displaying Data Using Equations to Display Messages   ÇR ÇH13-16Simple Programming  V OL  A RE A   Keys RPN mode R H V  R R H   V  S     13-18Simple Programming Displaying Information without Stopping  C  value Programming a Stop or Pause STOP, PSE Error StopsStopping or Interrupting a Program Interrupting a Running Program Editing a Program 13-20Simple Programming Program Memory Viewing Program Memory Memory Usage Catalog of Programs MEM13-22Simple Programming Clearing One or More Programs Checksum  Nonprogrammable FunctionsProgramming with Base   Selecting a Base Mode in a Program Decimal mode setNumbers Entered in Program Lines Polynomial Expressions and Horners Method 13-26Simple ProgrammingAx4 + Bx3 + Cx2 + Dx + E   A X     A  13-28Simple Programming Calling Subroutines XEQ, RTN Programming TechniquesRoutines in Programs Programming Techniques Nested Subroutines 14-2Programming Techniques A2 + b2 + c2 + d2 Branching GTO 14-4Programming TechniquesA2 + B2 Using GTO from the Keyboard Programmed GTO Instruction Conditional Instructions 14-6Programming Techniques Tests of Comparison x?y, x?0 Test Menus Program Lines Description RPN mode 14-8Programming Techniques Flags Meanings of Flags Flag Fraction-Control Flags Clear14-10Programming Techniques Programming Techniques Flags Menu 14-12Programming Techniques … takes the natural log of the X-input   14-14Programming TechniquesS  value   value Number Loops F     14-16Programming Techniques Conditional Loops GTO Loops with Counters DSE, ISG 14-18Programming Techniques Programming Techniques Indirectly Addressing Variables and Labels Variables I and J14-20Programming Techniques Indirect Address, I and J If I/J contains Then I/J will address 14-22Programming Techniques Program Control with I/J Equations with I/JUnnamed indirect variables 14-24Programming Techniques Solving and Integrating Programs Solving a ProgramSolving and Integrating Programs 15-2Solving and Integrating Programs V= N x R x T G Keys Display Description ALG modeProgram Lines Description ALG mode P  value   value H     L P   Using Solve in a Program 15-6Solving and Integrating Programs label  variable Program Lines In RPN mode Integrating a Program 15-8Solving and Integrating Programs  S   X Sin Using Integration in a Program 15-10Solving and Integrating Programs label   variable Restrictions on Solving and Integrating 15-12Solving and Integrating Programs Statistics Programs Curve FittingStatistics Programs = B + MIn 16-2Statistics Programs Program Listing 16-4Statistics Programs    16-6Statistics Programs Determines if D001 or B001 should be run 16-8Statistics Programs S             Logarithmic Exponential PowerNormal and Inverse-Normal Distributions  Qx = 0.5 − σ 12π xx e−x −x ⎟σ 2 ⎟2dx 16-12Statistics Programs This routine initializes the normal distribution program Program Lines 16-14Statistics Programs Program Lines To the inverse S 16-16Statistics Programs Program D  Grouped Standard Deviation 16-18Statistics Programs∑ x f   16-20Statistics Programs Index variable used to indirectly address the correct Statistics registerRegister Summation Σ f Summation Σ x i f    U     16-24Statistics Programs Miscellaneous Programs and Equations Miscellaneous Programs and EquationsTime Value of Money TVM equation is I P4 NÕ I 17-4Miscellaneous Programs and Equations KeysDisplay In RPN mode    P  value     I 17-6Miscellaneous Programs and Equations Prime Number Generator   17-8Miscellaneous Programs and Equations Tests to see whether all possible factors have been tried   17-10Miscellaneous Programs and Equations Cross Product in Vectors 17-12Miscellaneous Programs and Equations z 17-14Miscellaneous Programs and Equations Appendixes and Reference Page Calculator Support Support, Batteries, and ServiceSupport, Batteries, and Service A-1 Answers to Common Questions 2Support, Batteries, and Service Environmental Limits Support, Batteries, and Service A-3 Changing the Batteries Testing Calculator Operation 4Support, Batteries, and Service Support, Batteries, and Service A-5 Self-Test 6Support, Batteries, and Service Warranty Support, Batteries, and Service A-7 8Support, Batteries, and Service Customer Support Support, Batteries, and Service A-9 Emea 10Support, Batteries, and Service Support, Batteries, and Service A-11 Invent 12Support, Batteries, and Service Support, Batteries, and Service A-13 Yyyy+ 14Support, Batteries, and Service User Memory and the Stack Managing Calculator MemoryUser Memory and the Stack B-1 Resetting the Calculator 2User Memory and the Stack Clearing Memory User Memory and the Stack B-3 Category DefaultStatus of Stack Lift 4User Memory and the Stack Disabling Operations Neutral OperationsUser Memory and the Stack B-5 Status of the Last X Register ˆ ˆ6User Memory and the Stack Accessing Stack Register Contents 8User Memory and the Stack , ALG SummaryAbout ALG ALG Summary C-1 Doing Two argument Arithmetic in ALG Simple Arithmetic Power Functions Percentage Calculations Quotient and Remainder Of Division Parentheses CalculationsPermutations and Combinations xÕ  4 Õ  Trigonometric Functions Hyperbolic functions Reviewing the Stack ALG Summary C-7 Integrating an Equation Operations with Complex Numbers8ALG Summary 6 ALG Summary C-9  8Ë4 66 6Õ4 6 Entering Statistical Two-Variable Data z Σ Z612ALG Summary   ˆ Õ ˆ ALG Summary C-13 14ALG Summary More about Solving How Solve Finds a RootMore about Solving D-1 2More about Solving Function Whose Roots Can Be Found Interpreting Results More about Solving D-3 X X  X X  4More about Solving X  XX   XÕ 6More about Solving X Õ  X  X4 More about Solving D-7 When Solve Cannot Find a Root  X  8More about Solving  X  More about Solving D-9  X  X   10More about Solving X4 X ÕÕ More about Solving D-11 X   12More about Solving Round-Off Error  X  More about Solving D-13 14More about Solving More about Integration How the Integral Is EvaluatedMore about Integration Conditions That Could Cause Incorrect Results 2More about Integration ∞ xe−xdx   4More about Integration More about Integration 6More about Integration Conditions That Prolong Calculation Time   8More about Integration More about Integration 10More about Integration Messages Messages F-1 2Messages Or program. This might take a while Messages F-3 4Messages    Solve include EQN and PGM modecannot find FN each uses a level, they can also generate this error Messages F-5 6Messages Operation Index Name Keys and DescriptionOperation Index G-1 × Ø2Operation Index Operation Index G-3 = 4Operation Index Operation Index G-5  Denominator 6Operation Index label Operation Index G-7Natural exponential  Fractional part  8Operation Index Input variable variable Operation Index G-9  Integer part ISG variable variable LBL label label 10Operation Index Natural logarithm  Pause Operation Index G-11RCL variable Degrees to radians. Returns 2π/360 12Operation Index Operation Index G-13 View variable variable 14Operation IndexHyperbolic Tangent XEQ label label Õ≤ ÕÕ≠ Operation Index G-15 14-7 Menu ≠0? 16Operation Index=y? 11-4 Power Operation Index G-17 18Operation Index Index Index-1 Index-2 Index-3 EQN List TOP 6-7, F-2 Index-4 Index-5 Index-6 Index-7 Index-8 Index-9 Index-10 Index-11 Angle between two vectors Index-12