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HP 32SII More about Integration, How the Integral Is Evaluated, More about Integration D–1

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More about Integration

This appendix provides information about integration beyond that given in chapter 8.

How the Integral Is Evaluated

The algorithm used by the integration operation, ∫ Gº, calculates the integral of a function f(x) by computing a weighted average of the function's values at many values of x (known as sample points) within the interval of integration. The accuracy of the result of any such sampling process depends on the number of sample points considered: generally, the more sample points, the greater the accuracy, if f(x) could be evaluated at an infinite number of sample points, the algorithm could — neglecting the limitation imposed by the inaccuracy in the calculated function f(x) — always provide an exact answer.

Evaluating the function at an infinite number of sample points would take forever. However, this is not necessary since the maximum accuracy of the calculated integral is limited by the accuracy of the calculated function values. Using only a finite number of sample points, the algorithm can calculate an integral that is as accurate as is justified considering the inherent uncertainty in f(x).

The integration algorithm at first considers only a few sample points, yielding relatively inaccurate approximations. If these approximations are not yet as accurate as the accuracy of f(x) would permit, the algorithm is iterated (repeated) with a larger number of sample points. These iterations continue, using about twice as many sample points each time, until the resulting approximation is as accurate as is justified considering the inherent uncertainty in f(x).

More about Integration D–1

File name 32sii-Manual-E-0424
Printed Date : 2003/4/24	Size : 17.7 x 25.2 cm

Contents

HP 32SII RPN Scientific Calculator Owner’s Manual HP Part No. 00032–90068 Printed in Singapore Notice Part 1. Basic Operation 1.Getting Started 2. The Automatic Memory Stack 3. Storing Data into Variables 4. Real–NumberFunctions 5. Fractions 6.Entering and Evaluating Equations 7. Solving Equations 8.Integrating Equations Operations with Comb Numbers 10. Base Conversions and Arithmetic 11. Statistical Operations Part 2. Programming 12. Simple Programming 13. Programming Techniques 14. Solving and Integrating Programs Mathematics Programs Statistics Programs 17. Miscellaneous Programs and Equations Part 3. Appendixes and Regerence A. Support, Batteries, and Service B. User Memory and the Stack More about Solving More about Integration E.Messages F.Operation Index Index Page Basic Operation Page Getting Started Shifted Menu name function Letter for alphabetic key Keys for Clearing KeyDescription a Backspace Keyboard–entrymode: aclears the entire number 1 – 5 PICTURE HP 32II Menus Menu Chapter Name ºˆ ¸ˆ HP 32II Menus (continued) Other functions Example: Keys: Display: picture 1 – HP 32SII Annunciator Annunciator Meaning Upper Row: PRGM Keying in Numbers Getting Started 1–9 Making Numbers Negative Exponent of Ten Exponents in the Display Keys: Display:Description: notation because otherwise no Understanding Digit Entry Getting Started 1–11 Range Number and OVERFLOW Doing Arithmetic One–NumberFunctions 1–12Getting Started Two–NumberFunctions Note Getting Started 1–13 For example: To calculate: Controlling the Display Format Periods and Commas in Numbers 1–14Getting Started Number of Decimal Places Fixed–DecimalFormat ({%}) Scientific Format ({ }) Getting Started 1–15 Engineering Format ({}) SHOWing Full 12–DigitPrecision 1–16Getting Started Fractions Entering Fractions Getting Started 1–17 Keys: Display: 1–18Getting Started Displaying Fractions zŠ Messages Getting Started 1–19 Calculator Memory Checking Available Memory Clearing All of Memory To clear all of memory: 1–20Getting Started Getting Started 1–21 Page The Automatic Memory Stack The X–RegisterIs in the Display Clearing the X–Register z b 2–2The Automatic Memory Stack Reviewing the stack R¶ (Roll Down) Rµ (Roll Up) The Automatic Memory Stack 2–3 Exchanging the X– and Y–Registersin the Stack Arithmetic–Howthe Stack Does It 2–4The Automatic Memory Stack How ENTER Works The Automatic Memory Stack 2–5 Using a Number Twice in a Row Filling the to with a Constant 2–6The Automatic Memory Stack šY How CLEAR x Works The Automatic Memory Stack 2–7 The LAST X Register z Ž 2–8The Automatic Memory Stack Correcting Mistakes with LAST Wrong One–NumberFunction Mistakes with a Two–numberoperation The Automatic Memory Stack 2–9 zŽp Reusing Numbers with LAST 2–10The Automatic Memory Stack LAST The Automatic Memory Stack 2–11 Chain Calculations Work from the Parentheses Out 2–12The Automatic Memory Stack calculated 7 y )Pressing the function key produces the answer. This result can be used in further calculations Exercises Calculate: Solution: ™y ™ y<™ Order of Calculation The Automatic Memory Stack 2–15 More Exercises A Solution: Z„p 0„p 2–16The Automatic Memory Stack yzŽ y„p< The Automatic Memory Stack 2–17 Page Storing Data into Variables 3- 1 Picture Example: Storing Numbers Viewing a Variable without Recalling It 3–2Storing Data into Variables Reviewing Variables in the VAR Catalog To review the values at any or all non–zerovariables: Clearing Variables To clear a single variable: Storing Data into Variables 3–3 Arithmetic with Stored Variables Storage Arithmetic 3–4Storing Data into Variables Recall Arithmetic Kp Storing Data into Variables 3–5 H™D {‰D Exchanging x with Any Variable 3–6Storing Data into Variables {YA {Y The Variable "i Storing Data into Variables 3–7 Page Real–NumberFunctions To Calculate: Power Functions Result: _š3 z 4–2 Real–NumberFunctions _z Trigonometry Entering π Setting the Angular Mode Option RAD Trigonometric Functions Keys:Display:Description: MyQ 4–4 Real–NumberFunctions Programming Note: Hyperbolic Functions To Calculate z7N z7Q z7T Note The 4–6 Real–NumberFunctions Conversion Functions Coordinate Conversions To convert between rectangular and polar coordinates: Real–NumberFunctions 4–7 y, xθ, r θ,ry Example: Conversion with Vectors Time Conversions Real–NumberFunctions 4–9 To convert between decimal fractions and minutes–seconds: Example: Converting Time Formats Display:Description: Angle Conversions To convert an angle between degrees and radians: 4–10 Real–NumberFunctions Unit conversions To Convert: Probability Functions Factorial Gamma Real–NumberFunctions 4–11 Probability Menu PROB Menu Menu Label 4–12 Real–NumberFunctions Example: Combinations of People 24 š6 Twenty–fourpeople grouped six at a time Q8T Q8T Parts of Numbers PARTS Menu Names of Function 4–14 Real–NumberFunctions Fractions zŠ zŠ Fractions in the Display Display Rules 5–2Fractions Examples: Entered Value Accuracy Indicators Fractions 5–3 Tmeans the exact numerator is "a little below doesn't Longer Fractions 5–4Fractions 14 HA {‰A Changing the Fraction Display Setting the Maximum Denominator Fractions 5–5 Choosing Fraction Format Factors of denominator Fixed denominator To Get This Fraction Format: Change These Flags: Examples of Fraction Displays How 2.77 Is Displayed Format /c Factors of Rounding Fractions 5–8Fractions Fractions in Equations Fractions 5–9 Fractions in Programs 5–10Fractions Entering and Evaluating Equations y{My yKL 6–2Entering and Evaluating Equations Summary of Equation Operations Entering and Evaluating Equations 6–3 KeyOperation Entering Equations into the Equation List 6–4Entering and Evaluating Equations To enter an equation: Variables in Equations Number in Equations Entering and Evaluating Equations 6–5 Functions in Equations 6–6Entering and Evaluating Equations Parentheses in Equations Example: Entering an Equation KR {c /¾ Displaying and Selecting Equations Entering and Evaluating Equations 6–7 To display equations: To view a long equation: To select an equation: Example: Viewing an Equation 6–8Entering and Evaluating Equations Editing and Clearing Equations To edit an equation you're typing: To edit a saved equation: To clear an equation you're typing: Entering and Evaluating Equations 6–9 Types of Equations The HP 32SII works with three types of equations: Assignments assignment 6–10Entering and Evaluating Equations Evaluating Equations Entering and Evaluating Equations 6–11 Type of Equation Result for š Result for W Using ENTER for Evaluation 6–12Entering and Evaluating Equations Example: Evaluating an Equation with ENTER {G(z 35f `6 p Entering and Evaluating Equations 6–13 Using XEQ for Evaluation Example: Evaluating an Equation with XEQ Responding to Equation Prompts 6–14Entering and Evaluating Equations To leave the number unchanged, just press f The Syntax of Equations Operator Precedence Entering and Evaluating Equations 6–15 Order Operation Example Equations 6–16Entering and Evaluating Equations Equation Function Entering and Evaluating Equations 6–17 sin Parentheses used to group items Single No implied Division is done letter Verifying Equations 6–20Entering and Evaluating Equations Entering and Evaluating Equations 6–21 Example: Checksum and Length of an Equation Page Solving Equations Example: Solving the Equation of Linear Motion 7–2Solving Equations KV yK yKT {œT Solving Equations 7–3 Example: Solving the Ideal Gas Law Equation {GKP y º¾ KV {c KN y KR yKT Understanding and Controlling SOLVE Solving Equations 7–5 See appendix C for more information about how SOLVE works Verifying the Result approximation 7–6Solving Equations Interrupting a SOLVE Calculation Choosing Initial Guesses for SOLVE Solving Equations 7–7 Example. Using Guesses to Find a Root 7–8Solving Equations {G KV {c #/¾ KH {] Solving Equations 7–9 {œH 7–10Solving Equations For More Information Solving Equations 7–11 Page Integrating Equations I = ∫abf(x)dx Integrating Equations ( ∫ FN) To Integrating Equations: 8–2Integrating Equations J0 = π1 ∫0πcos(x sint)dt {Mp Keys:Description: 8–4Integrating Equations sinx 0 š2 first) Displays the current equation {)X Accuracy of Integration approximates might Specifying Accuracy accuracy Interpreting Accuracy Example: Specifying Accuracy Integrating Equations 8–7 Example: Changing the Accuracy 8–8Integrating Equations Integrating Equations 8–9 Page Operations with Comb Numbers x+ iy Real Stack Complex Stack 9–2Operations with Comb Numbers Complex Operations To do an operation with one complex number: Functions for One Complex Number, z Change sign,– zF To do an arithmetic operation with two complex numbers: Arithmetic With Two Complex Numbers, z1 and z2 zF™ zF„ zFy when calculating just one complex numbers Using Complex Number in Polar Notation imaginary (a, b) real Example: Vector Addition 185 lb 62 o 170 lb 143 o 100 lb261 o Page Base Conversions and Arithmetic Arithmetic in Bases 2, 8, and 10–2Base Conversions and Arithmetic Display: Description: Base Conversions and Arithmetic 10–3 The Representation of Numbers Negative Numbers 10–4Base Conversions and Arithmetic 546zw{%} Range of Numbers Range of Numbers for Base Conversions Base Positive Integer Negative Integer 10- 7B Picture {‰A Base Conversions and Arithmetic 10–7 Page Statistical Operations Entering One–VariableData 1.Press zb{Σ} to clear existing statistical data 2.Key in each x–valueand press 3.The display shows n, the number of statistical data values now accumulated To recall a value to the display immediately after it has been entered, press Correcting Errors in Data Entry Initial x, y Corrected x, y zb{´} Statistical Operations 11–3 Statistical Calculations Statistics Menus Mean 11–4Statistical Operations y–valuesas weights or frequencies. The weights can be integers or non–integers Example: Mean (One Variable) Calculate the mean of the times. (Treat all data as x–values.) Enters the first time Enters the remaining data; Sample Standard Deviation Example: Sample Standard Deviation 11–6Statistical Operations Population Standard Deviation Example: Population Standard Deviation Linear regression Statistical Operations 11–7 L.R. (Linear Regression) Menu {ºˆ } {¸ˆ } º ¸ Example: Curve Fitting ºˆ ¸ˆ T P E z,{P} z,{E} Statistical Operations 11–9 {,{¸ˆ } Limitations on Precision of Data 11–10Statistical Operations Normalizing Close, Large Numbers Effect of Deleted Data Summation Values and the Statistics Registers Summation Statistics Statistical Operations 11–11 Example: Viewing the Statistics Registers The Statistics Registers in Calculator Memory 11–12Statistical Operations Access to the Statistics Registers Statistics Registers Register Number Statistical Operations 11–13 Programming Simple Programming zUŒŒ {M y zUŒŒ Designing a Program 12–2Simple Programming Program Boundaries (LBL and RTN) Program Labels Program Line Numbers Program Returns Simple Programming 12–3 Using RPN and Equations in Programs Strengths of RPN Operations Strengths of Equations Data Input and Output 12–4Simple Programming Entering a Program To enter a program into memory: 1.Press zdto activate Program–entrymode program pointer Give the program a Keys That Clear 12–6Simple Programming Function Names in Programs Example: Entering a Labeled Program z“A Simple Programming 12–7 Example: Entering a Program with an Equation Running a Program 12–8Simple Programming z U Executing a Program (XEQ) run stop If necessary, enter the data before executing the program 5 WA z U Œ Œ Example: Testing a Program 12–10Simple Programming Entering and Displaying Data Using INPUT for Entering Data Simple Programming 12–11 "R" is the variable's name "?" is the prompt for information, and 0.0000 is the current value stored in the variable (run/stop) The area–of–a–circleprogram with an INPUT instruction looks like this: To respond to a prompt: To cancel the INPUT prompt To display digits hidden by the prompt, press {. (If Simple Programming 12–13 Using VIEW for Displaying Data Using Equations to Display Messages 12–14Simple Programming Example: INPUT, VIEW, and Messages in a Program zdz UŒŒ z“C zˆR zˆH {G{ MyKR 02 yKH Simple Programming 12–15 {G2 y{M yKR y {\KR ™KH { Displaying Information without Stopping Simple Programming 12–17 Stopping or Interrupting a Program Programming a Stop or Pause (STOP, PSE) Interrupting a Running Program Error Stops 12–18Simple Programming Editing Program To delete a program line: To insert a program line: To edit an equation in a program line: Simple Programming 12–19 Program Memory Viewing Program Memory UŒŒ Memory Usage 12–20Simple Programming The Catalog of Programs (MEM) z d Simple Programming 12–21 Clearing One or More Programs To clear a specific program from memory To clear all programs from memory: The Checksum 12–22Simple Programming Nonprogrammable Functions zb{}zUŒŒ azX z˜, z— { zd {G zŠ Programming with BASE Selecting a Base Mode in a Program Numbers Entered in Program Lines Decimal mode set: Hexadecimal mode set: 12–24Simple Programming Polynomial Expressions and Horner's Method zdz z“P zˆX Simple Programming 12–25 12–26Simple Programming Simple Programming 12–27 Page Programming Techniques Calling Subroutines (XEQ, RTN) 13–2Programming Techniques Nested Subroutines Programming Techniques 13–3 Example: A Nested Subroutine 246 13–4Programming Techniques Branching (GTO) A Programmed GTO Instruction Programming Techniques 13–5 Using GTO from the Keyboard 13–6Programming Techniques Conditional Instructions Programming Techniques 13–7 Tests of Comparison (x?y, x?0) The Test Menus Program Lines: 13–8Programming Techniques ! º<¸@ ! ! Flags set true false Testing a flag Flags 7, 8, and Clear Set 13–10Programming Techniques If the next program line is a PSE instruction, execution continues after a it doesn't affect automatic prompting during keyboard execution Flag 11 is automatically cleared after evaluation, SOLVE, or Annunciators for Set Flags Programming Techniques 13–11 Using Flags FLAGS Menu Menu Key Example: Using Flags 13–12Programming Techniques Clears flag 0, the indicator for In X. Clears flag 1, the indicator for In Y Sets flag 0, the indicator for In X. Clears flag 1, the indicator for In Y Clears flag 0, the indicator for In X. Sets flag 1, the indicator for In Y Sets flag 0, the indicator for ln X. Sets flag 1, the indicator for In Y If flag 0 is set Example: Controlling the Fraction Display 13–14Programming Techniques Programming Techniques 13–15 c + format (denominator is factor of 16), then shows the fraction f% Message indicates the fraction c + format (denominator is 16), then Loops Conditional Loops (GTO) 13–16Programming Techniques Program lines: Loops With Counters (DSE, ISG) Programming Techniques 13–17 The Loop–ControlNumber 13–18Programming Techniques Programming Techniques 13–19 Indirectly Addressing Variables and Labels The Variable "i 13–20Programming Techniques The Indirect Address, (i) indirectly If i contains: Then (i) will address: variable A or label A Program Control with (i) Example: Choosing Subroutines With (i) 13–22Programming Techniques If i hold: Then XEQ(i) calls: Example: Loop Control With (i) Programming Techniques 13–23 Equations with (i) 13–24Programming Techniques Programming Techniques 13–25 Page Solving and Integrating Programs Example: Program Using RPN 14–2Solving and Integrating Programs zdz {VG Solving and Integrating Programs 14–3 Example: Program Using Equation zdz z“H {x KP y Using SOLVE in Program Solving and Integrating Programs 14–5 Example: SOLVE in a Program 14–6Solving and Integrating Programs Integrating a Program To integrate a programmed function: Select the program that defines the function to integrate: press lower limit upper limit Si (t ) = ∫t ( sin x)dx 14–8Solving and Integrating Programs {VS Using Integration in a Program ∫ D e Solving and Integrating Programs 14–9 Restrictions o Solving and Integrating 14–10Solving and Integrating Programs Mathematics Programs Program Listing: 15–2Mathematics Programs Mathematics Programs 15–3 15–4Mathematics Programs Mathematics Programs 15–5 15–6Mathematics Programs Flags Used: Memory Required: Remarks: Program Instructions: Mathematics Programs 15–7 Variables Used: 15–8Mathematics Programs Transmitter Antenna Example 2: T = 80 o P = 74 o 15–10Mathematics Programs Mathematics Programs 15–11 Solutions of Simultaneous Equations  J   B E H   L  X  Mathematics Programs 15–13 15–14Mathematics Programs Mathematics Programs 15–15 15–16Mathematics Programs Mathematics Programs 15–17 15–18Mathematics Programs Mathematics Programs 15–19 Polynomial Root Finder 15–20Mathematics Programs Mathematics Programs 15–21 15–22Mathematics Programs Mathematics Programs 15–23 15–24Mathematics Programs Mathematics Programs 15–25 15–26Mathematics Programs Mathematics Programs 15–27 15–28Mathematics Programs Terms mid Coefficients Mathematics Programs 15–29 Example 1: 15–30Mathematics Programs Example 3: Coordinate Transformations Mathematics Programs 15–31 15–32Mathematics Programs Old coordinate system [0, 0] [m, n] New coordinate Program Lines:Description 15–34Mathematics Programs Mathematics Programs 15–35 Remark: 15–36Mathematics Programs (6, 8) 9, 7) (_5 (M, N) (2.7 15–38Mathematics Programs Statistics Programs A12 error will occur if a negative number is entered for these cases 16–2Statistics Programs Statistics Programs 16–3 16–4Statistics Programs Statistics Programs 16–5 16–6Statistics Programs Statistics Programs 16–7 Program instructions: 16–8Statistics Programs 104.5f 38.6f 102f Statistics Programs 16–9 16–10Statistics Programs Logarithmic Exponential Power Normal and Inverse–NormalDistributions Statistics Programs 16–11 Q(x) = 0.5 − σ 12π ∫xx e−((x−x)⎟σ )2 ⎟2dx 16–13 Checksum and length: F79E This subroutine calculates the integrand for the normal . ª º ª -+. H% ! Returns to the calling routine. Checksum and length: 3DC2 155.5 bytes: 107.5 for program, 48 for variables Statistics Programs 16–15 16–16Statistics Programs Statistics Programs 16–17 (∑xifi)2 (∑fi ) −1 ∑x f ∑x 2f Statistics Programs 16–19 16–20Statistics Programs Group Statistics Programs 16–21 16–22Statistics Programs Time Value of Money P1− (1+ I 100−N  + F(1+ (I 100))−N + B Miscellaneous Programs and Equations 17–1 17–2 Miscellaneous Programs and Equations y {]0 y SOLVE instructions: Miscellaneous Programs and Equations 17–3 17–4Miscellaneous Programs and Equations Part {œI zI 10™ 12y Miscellaneous Programs and Equations 17–5 Prime Number Generator 17–6Miscellaneous Programs and Equations Miscellaneous Programs and Equations 17–7 17–8Miscellaneous Programs and Equations Miscellaneous Programs and Equations 17–9 789 WP 17–10Miscellaneous Programs and Equations Appendixes and Reference Page Support, Batteries and Service z Ÿ Environmental Limits Support, Batteries, and Service A- 3 picture Testing Calculator Operation The Self–Test 1.Hold down the †key, then press 0, at the same time )8 Starting at the upper left corner 4.The self–testproduces one of these two results: Limited One–YearWarranty What Is Covered What Is Not Covered ANY OTHER IMPLIED WARRANTY OF MERCHANTABILITY OR FITNESS IS LIMITED TO THE ONE–YEAR If the Calculator Requires Service In the United States: Hewlett–PackardS.A In other countries: Support, Batteries, and Service A–7 Service Charge Shipping Instructions Warranty on Service Service Agreements Regulatory Information West Germany Noise Declaration Support, Batteries, and Service A–9 Page User Memory and the Stack Memory Requirements Data or Operation Amount of Memory Used B–2User Memory and the Stack Resetting the Calculator Clearing Memory User Memory and the Stack B–3 Category CLEAR ALL The Status of Stack Lift B–4User Memory and the Stack Disabling Operations Neutral Operations User Memory and the Stack B–5 The Status of the LAST X Register B–6User Memory and the Stack More about Solving C–2 More about Solving Interpreting Results Example: An Equation With One Root More about Solving C–3 ™4 y KX 02 „6 yKX ™8 š Example: An Equation with Two Roots C–4 KX 02 ™ KX „6 š%:-%. )Checksum and length Now, solve the equation to find its positive and negative roots: Your initial guesses for the Example: Discontinuous Function KX {]{ C–6 9{ Example: A Pole {G More about Solving C–7 {\KX When SOLVE Cannot Find Root More about Solving C–9 Example: A Relative Minimum {œX Example: An Asymptote C–10More about Solving 2 _{G More about Solving C–11 Example: A Math Error <KX p {\KX ™Œ3 { ]{]„ C–12More about Solving {‰X Example : A Local "Flat" Region More about Solving C–13 {VJ Round–OffError C–14More about Solving Underflow More about Solving C–15 Page More about Integration Conditions That Could Cause Incorrect Results More about Integration More about Integration D–3 ∫0∞ xe−xdx More about Integration D–5 D–6 More about Integration D–7 Conditions That Prolong Calculation Time More about Integration D–9 D–10More about Integration Messages ≥ 34or ≤ Messages E–3 Self–TestMessages: E–4Messages Operation Index F–2 {2{σ¸} {r z7zO F–3 z7zL {,{E} zw{} F–4 zb{º} F–5 F–6 F–7 zUŒ zw{%} zt zs (i) zŽ F–9 zw{} F–10Operation Index Operation Index F–11 F–12Operation Index z {,{ºˆ } Operation Index F–13 zl{≠} zl{≤} zl{≥} F–14Operation Index zn{≠} zn{≤} zn{≥} z,{¸ˆ } Notes: Page Index Index–2 cash flows, 17-1catalogs leaving, 1-3program, 1-21, 12-22using, 1-21variable, 1-21 1-14 checksums equations, 6-21, 12-7, 12-24programs, 12-22 Index–4 Index–5 Index–6 Index–7 Index–8 Index–9 Index–10 Index–11 Index–12 Index–13 Index–14 Index–15