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HP 33s More about Integration, How the Integral Is Evaluated, E–1, Gº, More about Integration

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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

E–1

Contents

user's guide Notice 1. Getting Started 2. RPN: The Automatic Memory Stack 3. Storing Data into Variables 4. Real–NumberFunctions 5. Fractions 6.Entering and Evaluating Equations Solving Equations Integrating Equations 9.Operations with Complex Numbers 10. Base Conversions and Arithmetic 11. Statistical Operations 12.Simple Programming Page 13. Programming Techniques 14. Solving and Integrating Programs 15. Mathematics Programs 16. Statistics Programs 17. Miscellaneous Programs and Equations A. Support, Batteries, and Service B. User Memory and the Stack C. ALG: Summary D. More about Solving E. More about Integration F. Messages G. Operation Index Index Page Basic Operation Page Getting Started Turning the Calculator On and Off Adjusting Display Contrast Shifted Keys Alpha Keys Left-shifted Right-shifted function Letter for Silver Paint Keys Backspacing and Clearing Keys for Clearing Key Getting Started 1–5 1–5 Keys for Clearing (continued) 1–6 Getting Started 1–6 Using Menus HP 33s Menus (continued) Description Other functions To use a menu function: 1–8 Exiting Menus RPN and ALG Keys The Display and Annunciators HP 33s Annunciators Annunciator Meaning EQN 1–12 HP 33s Annunciators (continued) Getting Started 1–13 1–13 Making Numbers Negative Exponents of Ten Understanding Digit Entry Range of Numbers and OVERFLOW One–NumberFunctions Two–NumberFunctions Periods and Commas in Numbers Number of Decimal Places SHOWing Full 12–DigitPrecision Entering Fractions 1–22 Getting Started 1–22 Displaying Fractions Checking Available Memory Clearing All of Memory RPN: The Automatic Memory Stack The X and Y–Registersare in the Display Clearing the X–Register Reviewing the Stack Exchanging the X– and Y–Registersin the Stack How ENTER Works How CLEAR x Works The LAST X Register RPN: The Automatic Memory Stack 2–7 2–7 Correcting Mistakes with LAST Reusing Numbers with LAST LAST 96.704 ‘ Enters first number 52.3947 › 2–10 Work from the Parentheses Out sequentially–entered Calculate 2 ⎟ (3 + 10): ‘10 › Calculate 4 ⎟ [14 + (7 ⋅ 3) – 2] : ‘3 z 14 ›2 Puts 4 before 33 in preparation for division Calculates 4 ⎟ 33, the answer 3 ‘4 › Exercises Order of Calculation More Exercises A Solution: […q# z{ z…q# RPN: The Automatic Memory Stack 2–15 2–15 Page Storing Data into Variables Storing and Recalling Numbers To store a copy of a displayed number (X–register)to a variable: To recall a copy of a number from a variable to the display: Example: Storing Numbers 3–2 Storing Data into Variables Viewing a Variable without Recalling It Reviewing Variables in the VAR Catalog To review the values at any or all non–zerovariables: Storing Data into Variables 3–3 3–3 Storage Arithmetic Recall Arithmetic I›E |ŠD |ŠE |ŠF Exchanging x with Any Variable | Z 3–6 Storing Data into Variables 3–6 |ZA The Variable "i other indirect addressing Storing Data into Variables 3–7 3–7 Page Real–NumberFunctions To Calculate: Quotient and Remainder of Division | D Power Functions 4–2 Real–NumberFunctions 4–2 Entering π Setting the Angular Mode Trigonometric Functions |NzR Programming Note: Real–NumberFunctions 4–5 4–5 Hyperbolic Functions {O {R {U {{M 4–7 Physics Constants CONST Menu Items Value 4–8 Real–NumberFunctions Conversion Functions Real–NumberFunctions 4–9 4–9 Coordinate Conversions Example: Polar to Rectangular Conversion 30o Example: Conversion with Vectors Real–NumberFunctions 4–11 4–11 Time Conversions Angle Conversions Unit Conversions Factorial Gamma Probability Example: Combinations of People Display: 4–15 Parts of Numbers Integer part Fractional part Absolute value Sign value Names of Functions usually in equations also) Real–NumberFunctions 4–17 4–17 Page Fractions Display Rules Accuracy Indicators Longer Fractions Setting the Maximum Denominator Choosing a Fraction Format Examples of Fraction Displays Number Entered and Fraction Displayed Format ¼ denominator Fixed Rounding Fractions Fractions 5–7 5–7 16 |Œ 56 3 4 ID Fractions in Equations See chapter 6 for information about working with equations 5–8 Fractions 5–8 Fractions in Programs Fractions 5–9 5–9 Page Entering and Evaluating Equations z|Nz zLL 6–2 Entering and Evaluating Equations 6–2 Summary of Equation Operations equation list Operation |H Enters and leaves Equation mode replacing "=" with "–"if an "=" is present Variables in Equations Numbers in Equations Functions in Equations Parentheses in Equations To view a long equation: To select an equation: Example: Viewing an Equation Editing and Clearing Equations 6–7 To edit an equation you're typing: To edit a saved equation: To clear an equation you're typing: 6–8 |H /ºº 1!.2 1 bº 1!.2-¾ bb/ºº 1!.2¾ Types of Equations The HP 33s works with three types of equations: Equalities + y = r 6–10 Using ENTER for Evaluation Using XEQ for Evaluation Responding to Equation Prompts Operator Precedence Order Operation Example Equations 6–14 Equation Functions sinθ Parentheses used to group items sin(π/n) Syntax Errors Solving Equations „If the displayed value is the one you want, press g If you want a different value, type or calculate the value and press You can halt a running calculation by pressing ‡or g Example: Solving the Equation of Linear Motion LD |dLV Starts the equation zLT › /#º!-¾ .5 zLG zLT 7–3 Example: Solving the Ideal Gas Law Equation P ⋅ V = N ⋅ R ⋅ T Enter the equation: |HLP z º¾ LV |d LN z LR zLT Understanding and Controlling SOLVE Solving Equations 7–5 7–5 Verifying the Result Interrupting a SOLVE Calculation Choosing Initial Guesses for SOLVE Example: Using Guesses to Find a Root 40 40_2H 7–8 Solving Equations 7–8 |H LV |d LH |` z|] z4 zLH 7–10 Solving Equations 7–11 7–11 Page Integrating Equations I = ≥ab f (x) dx Integrating Equations ( ≥ FN) To integrate an equation: lower Display the equation: Press Select the variable of integration: Press RLX |Nq ≥0π f (t ) Integrating Equations 8–3 8–3 sinx The current equation or ! ! OLX 1%¾ Accuracy of Integration might Integrating Equations 8–5 8–5 Specifying Accuracy Interpreting Accuracy 8–7 its uncertainty indicates 8–8 Integrating Equations 8–8 Operations with Complex Numbers Complex function (displayed) imaginary part Complex Operations To do an operation with one complex number: 9–2 Operations with Complex Numbers 9–2 Functions for One Complex Number, z {G^ {G {G {GO {GR {GO 1 ‘2 ^‘ 3 ^‘4 { {G {Gz Using Complex Numbers in Polar Notation 9–5 Example: Vector Addition 9–6 Operations with Complex Numbers 9–6 9–7 Page Base Conversions and Arithmetic Arithmetic in Bases 2, 8, and 10–2 Base Conversions and Arithmetic 10–2 Base Conversions and Arithmetic 10–3 10–3 Negative Numbers Range of Numbers Windows for Long Binary Numbers Statistical Operations Entering One–VariableData Entering Two–VariableData Correcting Errors in Data Entry Reenter the incorrect data, but instead of pressing 2.Enter the correct value(s) using 11–3 Mean Example: Weighted Mean (Two Variables) Price per Part (x) Number of Parts (y) Statistical Operations 11–5 11–5 Sample Standard Deviation Population Standard Deviation Linear Regression Example: Curve Fitting X, Nitrogen Applied Y, Grain Yield | {T} º ¸ 11–8 Statistical Operations 11–8 (70, y) Limitations on Precision of Data Statistical Operations 11–9 11–9 Summation Statistics The Statistics Registers in Calculator Memory Access to the Statistics Registers Statistics Registers Register Number 11–12 Programming Page Simple Programming RPN mode Keys: (In RPN mode) {V 12–2 Selecting a Mode Program Boundaries (LBL and RTN) Using RPN, ALG and Equations in Programs Data Input and Output These are covered later in this chapter under "Entering and Displaying Data Entering a Program { e To enter a program into memory: 1.Press {eto activate Program–entrymode program pointer Keys That Clear Function Names in Programs Example: Entering a Program with an Equation {e V {•E |H|N zLR Executing a Program (XEQ) Testing a Program {V Example: Testing a Program 12–10 Simple Programming 12–10 Using INPUT for Entering Data The area–of–a–circleprogram with an INPUT instruction looks like this: RPN mode ALG mode º º π Using VIEW for Displaying Data Using Equations to Display Messages {•C {‰R {‰H |H| NzLR Displaying Information without Stopping Programming a Stop or Pause (STOP, PSE) Interrupting a Running Program Error Stops Editing a Program To delete a program line: 12–18 Viewing Program Memory Memory Usage The Catalog of Programs (MEM) Clearing One or More Programs The Checksum Selecting a Base Mode in a Program Numbers Entered in Program Lines {‰X (In ALG mode) 12–24 Simple Programming 12–24 Simple Programming 12–25 12–25 Page Programming Techniques Calling Subroutines (XEQ, RTN) Nested Subroutines In RPN mode Starts subroutine here Enters A Enters B Branching (GTO) branching label 13–4 Programming Techniques 13–4 A Programmed GTO Instruction Using GTO from the Keyboard Conditional Instructions 13–6 Programming Techniques 13–6 Tests of Comparison (x?y, x?0) Flags Flags 0, 1, 2, 3, and Flag 5 overflow Flag 13–9 Flag 13–10 Programming Techniques 13–10 Annunciators for Set Flags Using Flags FLAGS Menu Programming Techniques 13–11 13–11 Example: Using Flags 13–12 Programming Techniques 13–12 Program Lines: Programming Techniques 13–13 13–13 Example: Controlling the Fraction Display 13–14 Programming Techniques 13–14 Programming Techniques 13–15 13–15 Loops 13–16 Programming Techniques 13–16 Conditional Loops (GTO) Loops with Counters (DSE, ISG) Programming Techniques 13–19 13–19 The Variable "i The Indirect Address, (i) Program Control with (i) & !- L & %1L2 If i holds: Then XEQ(i) calls: Compute 13–23 Equations with (i) Program Lines: Programming Techniques 13–25 13–25 Page Solving and Integrating Programs unknown separate every 3.Enter the instructions to evaluate the function 14–2 {e 14–3 |WG Example: Program Using Equation {•H LP z LV |’ |WH g@ g@ g!@ ‘10 !@ ) Using SOLVE in a Program Example: SOLVE in a Program 14–6 Solving and Integrating Programs 14–6 Program Lines: (In RPN mode) Integrating a Program To integrate a programmed function: Solving and Integrating Programs 14–7 14–7 | W To write a program for ≥ FN: 14–8 Solving and Integrating Programs 14–8 Si(t) = ≥ )dx ≥ G variable Example: ≥ FN in a Program 1 D 14–10 Restrictions on Solving and Integrating will result in an ≥ 1 ≥ 2 error. Also, SOLVE and contains an /label instruction; if attempted a # !# or ≥ !# error will be returned. SOLVE cannot call a routine that contains an Page Mathematics Programs v1 = X i + Y j + Z k and v2=U i + V j + W k 15–2 Mathematics Programs 15–2 (X2 + Y2 + Z2 ) and P 15–4 Mathematics Programs 15–4 Mathematics Programs 15–5 15–5 15–6 Mathematics Programs 15–6 Program Lines: (In ALG mode) Flags Used: Remarks: Program Instructions: 15–7 Variables Used: Example 1: 15–8 Mathematics Programs 15–8 N (y) Transmitter Antenna E (x) XR ^g g !@ Example 2: Mathematics Programs 15–9 15–10 Enters resultant vector 1.07g !@ ) 125 g 63 g 15–11 Solutions of Simultaneous Equations 15–12 Mathematics Programs 15–12 Mathematics Programs 15–13 15–13 15–14 Mathematics Programs 15–14 Mathematics Programs 15–15 15–15 Checksum and length: 7F00 º65¸ L L L º1L2 This routine multiples and adds values within a row. Gets next column value Sets index value to point to next row value 15–16 Mathematics Programs 15–17 15–17 15–18 Mathematics Programs 15–18 Mathematics Programs 15–19 15–19 g@ g@ ) g@ Inverts inverse to produce original matrix Polynomial Root Finder For this program, a general polynomial has the form xn + an–1xn–1 + ... + a1x + a0 y3 + b2y2 + b1y + b0 where b2 = – a2 15–21 15–22 Mathematics Programs 15–22 ª Gets synthetic division coefficients for next lower order polynomial Generates DIVIDE BY 0 error if no real root found. Checksum and length: 15FE Starts quadratic solution routine. Exchanges a0 and a1 15–23 15–24 Mathematics Programs 15–24 Mathematics Programs 15–25 15–25 15–26 Mathematics Programs 15–26 ! !ª º Stores 1 or JK – a1/2 Calculates sign of C J2 -– a2 15–27 " " " ! L " #$ L " #$ % " L " ! L " #$ L Displays complex roots if any 15–28 1.Press {c{} to clear all programs and variables 2.Key in the program routines; press ‡when done 3.Press XP to start the polynomial root finder 4.Key in F, the order of the polynomial, and press g 15–29 15–30 Mathematics Programs 15–30 Mathematics Programs 15–31 15–31 Example 3: Find the roots of the following quadratic polynomial: x2 + x – 6 1 g Coordinate Transformations This program provides two–dimensionalcoordinate translation and rotation u= (x – m) cosθ + (y – n) sinθ v = (y – n) cos θ –(x – m) sinθ The inverse transformation is accomplished with the formulas below x= u cosθ – v sinθ + m y = u sinθ + v cosθ + n 15–33 15–34 Mathematics Programs 15–34 ¸8º %º ! % º65¸ ! & º65¸ #$ % #$ & ! Pushes up results and recalls M Completes calculation by adding M and N to previous results 15–35 Remark: 15–36 Mathematics Programs 15–36 Mathematics Programs 15–37 15–37 15–38 Mathematics Programs 15–38 Statistics Programs y = Be Mx y = B + Mx y = BxM y = B + MIn Checksum and length: E3F5 16–3 16–4 Statistics Programs 16–4 Stores b in B Displays value Calculates coefficient m Stores m in M 16–5 16–6 Statistics Programs 16–6 Program instructions: Statistics Programs 16–7 16–7 16–8 Statistics Programs 16–8 Statistics Programs 16–9 16–9 16–10 Statistics Programs 16–10 Q(x) = 0.5 − σ 12π ≥xx e−((x −x )⎟σ )2 ⎟2dx This routine initializes the normal distribution program Stores default value for mean Prompts for and stores mean, M Stores default value for standard deviation 16–12 ! !- % ! ! ) ! º6¸@ ! ! ! Adds the correction to yield a new Xguess Loops to calculate another Checksum and length: 0E12 16–13 16–14 Statistics Programs 16–14 6.To calculate Q(X) given X, XD After the prompt, key in the value of 9.To calculate X given Q(X), press DDummy variable of integration 16–15 16–16 Statistics Programs 16–16 Grouped Standard Deviation ( ƒ x f )2 ) − Statistics Programs 16–17 16–17 16–18 Statistics Programs 16–19 16–19 16–20 Statistics Programs 16–20 Group Statistics Programs 16–21 16–21 16–22 Statistics Programs 16–22 Time Value of Money The TVM equation is: + F(1+ (I 100))−N + B Balance, B Payments, P z|]1 |]1 › LN |` qLI ›LF z |]1 ›LI 17–3 17–4 Miscellaneous Programs and Equations 17–4 Miscellaneous Programs and Equations 17–5 17–5 24 g .8) Retains P; prompts for Retains 0.56 in I; prompts for N Prime Number Generator 17–6 Miscellaneous Programs and Equations 17–6 Miscellaneous Programs and Equations 17–7 17–7 17–8 Miscellaneous Programs and Equations 17–8 Miscellaneous Programs and Equations 17–9 17–9 789 XP after 17–10 Miscellaneous Programs and Equations 17–10 Appendixes and Reference Page Support, Batteries and Service Answers to Common Questions Environmental Limits Changing the Batteries A–2 Support, Batteries, and Service A–2 To install batteries: A–3 Warning Do not mutilate, puncture, or dispose of batteries in fire. The batteries can burst or explode releasing hazardous chemicals Testing Calculator Operation The calculator won't turn on (steps If these steps fail to restore calculator operation, it requires service A–4 Support, Batteries, and Service A–4 The Self–Test 1.Hold down the ‡key, then press at the same time C™Ÿ˜—š 4.The self–testproduces one of these two results: „The calculator displays . if it passed the self–test.Go to step Warranty HP 33s Scientific Calculator; Warranty period: 12 months HP warrants to you, the A–6 Support, Batteries, and Service A–6 Service Europe Country : Telephone numbers Austria A–8 N.America Regulatory Information USA Canada Support, Batteries, and Service A–9 A–9 A–10 User Memory and the Stack Resetting the Calculator B–2 User Memory and the Stack B–2 Clearing Memory 1.Press and hold down the ‡key 2.Press and hold down Category CLEAR ALL Disabling Operations Neutral Operations V V B–5 The Status of the LAST X Register B–6 User Memory and the Stack B–6 ALG: Summary Simple Arithmetic Power Functions Percentage Calculations Permutations and Combinations Quotient and Remainder Of Division ¯ºy º y ALG: Summary C–5 C–5 Reviewing the Stack C–6 ALG: Summary C–6 Coordinate Conversions If x = 5, y = 30, what are r, θ 30 [5 {r 8´θ8T T/) 8´θ8T Integrating an Equation C–8 ALG: Summary C–8 Operations with Complex Numbers To enter a complex number To view the result of complex operations Complex Operations ALG: Summary C–9 G|` {G|` q|]2 ^› C–10 ALG: Summary C–10 Arithmetic in Bases 2, 8, and Entering Statistical Two–VariableData {c{´} ALG: Summary C–13 C–13 Page More about Solving f (x) Function Whose Roots Can Be Found D–2 More about Solving D–2 Interpreting Results „If it finds an estimate for which f(x) equals zero. (See figure a, below.) Cases Where a Root Is Found To obtain additional information about the result, press again to see the 2 ^z ›4 z LX 2 6 zLX ›8 ‘ D–5 Special Case: A Discontinuity and a Pole Example: Discontinuous Function |"LX | D–6 More about Solving D–6 x− 1 LX q |]LX More about Solving D–7 D–7 When SOLVE Cannot Find a Root D–8 Case Where No Root Is Found Example: A Relative Minimum 6 zLX › More about Solving D–9 D–9 b| Example: An Asymptote D–10 More about Solving D–10 ^IX ^|H Example: Find the root of the equation [x ⎟ (x + 0.3)] − 0.5 #LX q| |ŠX Example: A Local "Flat" Region D–12 More about Solving D–12 a8 ^IX Round–OffError More about Solving D–13 D–13 Underflow D–14 More about Solving D–14 More about Integration Conditions That Could Cause Incorrect Results whose E–2 More about Integration E–2 For example, consider the approximation of ≥0∞ xe −x dx Try it and see what happens. Enter the function f(x) = xe–x E–3 |H LX |` E–4 E–5 Calculated integral of this function will be accurate Calculated integral of this function may be inaccurate E–6 More about Integration E–6 Conditions That Prolong Calculation Time Rerun the previous integration problem with this new limit of integration: 0 ‘a3 _ New upper limit E–8 E–9 Page Messages F–2 Messages F–2 #1 #2 #1 ≥ 2 „the program that you called referred to another label, which does not exist The catalog of programs ( {Y{} ) indicates no program labels stored F–3 !12 Attempted to calculate the square root of a negative number Statistics error: „Attempted to do a statistics calculation with n F–4 Operation Index ˜or — {j {h G–2 Operation Index G–2 ƒ(xi − x)2 ⎟ n {{P G–4 Operation Index G–4 {{M G–5 {c{º} G–6 G–7 G–8 Operation Index G–8 Operation Index G–9 G–9 |u {t G–10 Operation Index G–10 (i) L”I” Operation Index G–11 G–11 G–12 Operation Index G–12 ƒ(xi − x )(yi − y ) ƒ(xi − x )2 ⋅ (yi − y )2 Operation Index G–13 G–13 G–14 Operation Index G–14 Operation Index G–15 G–15 ƒ(xi − x )2 ⎟ (n − 1) ƒ(yi − y )2 ⎟ (n − 1) Operation Index G–17 G–17 {n{≠} {n{≤} {n{≥} G–18 Operation Index G–18 |o{≠} |o{≤} |o{≥} Operation Index G–19 G–19 G–20 Index Index–2 program, 1–24, 12–20using, 1–24variable, 1–24, 3–3 1–17, 9–3checksums equations, 6–18, 12–6, 12–21programs, 12–20 CLEAR menu, 1–6clearing equations, 6–8 display format affects integration, 8–2, 8–5, 8–7 affects numbers, 1–19affects rounding, 4–16default, B–3 periods and commas in, 1–18, A–1 clearing stack, 2–5 Index–5 Index–6 imaginary part (complex numbers), 9–1, 9–2 indirect addressing, 13–20, 13–21, 13–22 integer–partfunction, 4–16integration in programs, 14–9interrupting, B–2 limits of, 8–2, 14–8, C–8, E–7memory usage, 8–2, B–2purpose, 8–1 Index–8 internal representation, 1–19, 10–4 large and small, 1–14, 1–16limitations, 1–14mantissa, 1–15 A–1 precision, 1–19, D–13prime, 17–6 typing, 1–14, 1–15, 10–1 moving to, 12–10, 12–19purpose, 12–3 typing name, 1–3viewing, 12–20 ALG operations, 12–4base mode, 12–22branching, 13–2, 13–4, 13–6 13–11, 13–17, 14–6 data input, 12–4, 12–11, 12–13data output, 12–4, 12–13 testing, 12–9 using integration, 14–9using SOLVE, 14–6 variables in, 12–11, 14–1, 14–7prompts affect stack, 6–13, 12–12clearing, 1–5, 6–13, 12–13equations, 6–12 INPUT, 12–11, 12–13, 14–2, 14–8 SOLVE, D–13statistics, 11–9trig functions, 4–4 routines calling, 13–2 nesting, 13–3, 14–11parts of programs, 13–1 RPN origins, 2–1running programs, 12–9 registers statistics calculating, 11–4 data accessing, 11–11 clearing, 1–6, 11–2, 11–11contain summations, 11–1 Index–14 Index–15