4040, 1416 7652 | HP 15c Scientific instruction

196 Section 14: Numerical Integration

Keystrokes	Display
¥
0 v	0.0000
$	3.1416
R	3.1416

Run mode.

Key lower limit, 0, into Y- register.

Key upper limit, π, into X- register.

Specify Radians mode for trigonometric functions.

Now you are ready to press ´f 0 to calculate the integral. When you do so, you'll find that – just as with _ – the calculator will not display the result right away, as it does with other operations. The HP-15C calculates integrals using a sophisticated iterative algorithm. Briefly, this algorithm evaluates f(x), the function to be integrated, at many values of x between the limits of integration. At each of these values, the calculator evaluates the function by executing the subroutine you write for that purpose. When the calculator must execute the subroutine many times – as it does when you press f– you can't expect any answer right away. Most integrals will require on the order of 2 to 10 seconds; but some integrals will require even more. Later on we'll discuss how you can decrease the time somewhat; but for now press ´f 0 and take a break (or read ahead) while the HP-15C takes care of the drudgery for you.

Keystrokes	Display
´f 0	2.4040	= ∫π cos (sinθ) dθ .
		0

In general, don't forget to multiply the value of the integral by whatever constants, if any, are outside the integral. In this particular problem, we need to multiply the integral by 1/ π to get J0 (1):

Keystrokes	Display
$	3.1416
÷	0.7652	J0 (1).

Image 196

HP 15c Scientific manual 4040, 1416 7652

Contents

HP-15C Owner’s Handbook HP Part Number 00015-90001 Edition 2.4, Sep Legal Notice Introduction Contents Contents Display and Continuous Memory Program Editing Program Branching and Controls Subroutines Calculating With Complex Numbers Indirect Display ControlCalculating With Matrices Numerical Integration Contents Appendix a Error Conditions Appendix C Memory AllocationAppendix D a Detailed Look at Appendix E a Detailed Look at f Contents Appendix F Batteries Function Summary and IndexProgramming Summary and Index Subject Index HP-15C Problem Solver Quick Look at Manual Solutions To Compute Keystrokes Display Programmed Solutions Keystrokes Display KeystrokesDisplay 001-42,21,11002 003 004 005 006 007 008 009 8313 300.51 HP-15C a Problem Solver Part l HP-15C Fundamentals Power On and Off Getting StartedKeyboard Operation Section Prefix Keys Changing SignsKeying in Exponents I O m ´ P I l F T s ? t H b Clear Keys Digit entry not terminated Display Clearing ` and −Clears only the last digit Calculations One-Number FunctionsTwo-Number Functions 6532 17 + 26.0000 22.0000 500013.0000 78.0000 Numeric Functions Number Alteration Functions One-Number Functions General Functions Time and Angle Conversions Trigonometric OperationsPressing Calculates Degrees/Radians Conversions 7069Radians 40.5000 Logarithmic Functions Hyperbolic Functions Power Function Two-Number FunctionsPercentages To Calculate Keystrokes Display Polar and Rectangular Coordinate Conversions Enters the base number the priceCalculates 3% of $15.76 the tax Polar Conversion. Pressing Keystrokes Display Automatic Memory Stack Last X, and Data Storage Automatic Memory Stack Stack ManipulationAutomatic Memory Stack Registers Always displayed Memory Stack, Last X, and Data Storage Stack Manipulation FunctionsLost Lost Last X Register and K 287.000022.2481 12.9000 13.9 + Calculator Functions and the Stack20.6475 Order of Entry and the v Key +15 X15 Nested Calculations 7 +65.0000 69.0000 Arithmetic Calculations With Constants 5 ‛15 Keys 1000 Keystrokes Display Growth factor000 Storage Register Operations Storing and Recalling Numbers322.5000 520.8750 Clearing Data Storage Registers Storage and Recall Arithmetic For storage arithmetic For recall arithmetic Problems Overflow and Underflow24 l-0 15.0000 Memory Stack, Last X, and Data Storage Probability Calculations Statistics Functions60.0000 Random Number Generator 270,725.00005764 3422 Accumulating Statistics Registers Register Contents 20.00 40.00 60.00 80.00 Kg per hectare Metric tons per Hectare, y20 z 61v 40 z 7.21 60 z 7.78 80 z l Σy2 Correcting Accumulated Statistics 20 w 20 z Standard Deviation Mean40.00 Linear Regression 31.62Standard deviation about the mean nitrogen Application Linear Estimation and Correlation Coefficient Statistics Functions Other Applications 70 ´j Display Control Display Continuous MemoryFixed Decimal Display Scientific Notation Display Engineering Notation Display234568 234567 Round-Off Error Special DisplaysMantissa Display Annunciators Error Display Digit Separators12,345.67 12.345.6700 Continuous Memory Low-Power IndicationStatus Resetting Continuous Memory Page Part ll HP-15C Programming Programming Basics MechanicsCreating a Program Loading a Program Programming Basics ´b a Running a Program Intermediate Program Stops002 003 004 005 006 007 008 How to Enter Data 300.51 300.51 ´A Program Memory Radius, r Height, h Base Area Volume Surface Area Totals 002 004 005007-44,40 010 Or G a Program Instructions Further InformationInstruction Coding Memory Configuration Keycode 25 second row, fifth key Initial Memory Configuration 60 ´ m% Program Boundaries ´ m %19 ´ m% 19.0000 Abbreviated Key Sequences Unexpected Program Stops´bA ´b3 End of memory User Mode Polynomial Expressions and Horners Method¤ @ y ∕ LOG % Nonprogrammable Functions 001-42,21,12002 003 004 005 006 007 008 009 0000 12,691.0000 Problems Program Editing Moving to a Line in Program Memory Deleting Program Lines ExamplesInserting Program Lines Or use Â Single-Step Operations Line Position ÂholdRelease Result Insertions and Deletions Initializing Calculator Status + i n InterestPV 1 + i n 100 270 ´bA D ´4 O0 2* O1 2÷ * ´ ´ l0 l1 ´r * n Program Branching Controls Branching Conditional Tests Test Flags  n will clear flag number n Example Branching and Looping 010-45,20 013-43,30014 016-44,40 Example Flags Formula is 002-43 004-42,21,15005-43, 4 006-42,21 Go to 250.000048.0000 10,698.3049 Looping Conditional Branching System Flags Flags 8 Program Branching and Controls Subroutines Go To Subroutine and ReturnSubroutine Execution ´b.1 Subroutine Limits 000 001- ´b9 002- R003- O0 004 ´ b.4 ´b.5 Subroutine Return Nested Subroutines V and % Keys Index Register Loop Control106 Indirect Program Control With the Index Register Program Loop ControlIndex Register Storage and Recall Index Register and Loop Control Exchanging the X-Register Index Register ArithmeticIndirect Branching With Indirect Display Format Control With Indirect Flag Control WithLoop Control With Counters I and e Nnnnn x x x y y 5 0 0 Start count at zero Count by twos Count up to Examples Register Operations IterationsStoring and Recalling Keystrokes Display 12.3456 Exchanging the X-Register Example Loop Control with eStorage Register Arithmetic Loop control number in R2 −− 011- 42012-42, 5 013- 22 15 O Example Display Format Control64.8420 0000 50.0000 Index Register Contents Indirect Display Control Index Register and Loop Control 118 Part lll HP-15C Advanced Functions Complex Stack and Complex Mode Calculating With Complex NumbersCreating the Complex Stack 120 Complex Numbers and the Stack Deactivating Complex ModeEntering Complex Numbers ´ % hold 8.0000 release Z 8 Y 7 X Keys Stack Lift in Complex Mode Manipulating the Real and Imaginary Stacks Clearing a Complex Number Or other operation Continue with any operation − 4 v Continue with any operation ´ %hold release Entering Complex Numbers with −. The clearing functions −0000 17.0000 144.0000 Entering a Real Number Followed by another number Entering a Pure Imaginary Number ´ Continue with any operation Operations With Complex Numbers Storing and Recalling Complex Numbers´ O L 2 ´ ¤x N o ∕ @ a + * ÷ y 2000 70000428 0491 Polar and Rectangular Coordinate Conversions Complex Results from Real Numbers5708 ´ % hold Release1.5708 Cos θ + i sin θ = re iθ Polar + ib = ∠ θ 2981 8452+ 3.1434 352.0000 872.00002361 4721 For Further Information Calculating With Matrices 138 Keystrokes Display Deactivates Complex Mode = A-1B Matrix Dimensions Running11.2887 2496 Dimensioning a Matrix Number Rows Columns Displaying Matrix Dimensions Changing Matrix Dimensions´mA Keystrokes l B Display Storing and Recalling Matrix Elements Storing and Recalling All Elements in Order ⎡ a Checking and Changing Matrix Elements Individually Keystrokes Display Storing a Number in All Elements of a Matrix Matrix OperationsMatrix Descriptors Result Matrix Copying a Matrix One-Matrix Operations Calculating with Matrices Scalar Operations LB b Elements of Result Matrix LA a Keystrokes Display Subtracts 1 from the elements Arithmetic OperationsLB b 2 LA a 2 Matrix Multiplication = AT B Keystrokes Display l a a Solving the Equation AX = B 24 OA 240086 OA 8600 274 OB 233 OB 331 OB 120.32 OB 112.96 OB 151.36 OB ´Á Calculating the Residual Week Cabbage kg 186 141 215 Broccoli kg 116 Using Matrices in LU Form Calculations With Complex Matrices Storing the Elements of a Complex Matrix Then Z can be represented in the calculator by Pressing Transforms Into = ⎢ LA a Complex Transformations Between ZP and Z Inverting a Complex Matrix Multiplying Complex Matrices ´ a Keystrokes lA lB Display Displays descriptor of matrix a ´U lC LC lC lC lC lC lC lC ´U Solving the Complex Equation AX = B ZZ −1 AX = B 200.0000 170.0000 0372 13110437 1543 Calculating with Matrices Using Matrix Descriptors in the Index Register Using a Matrix Element With Register OperationsMiscellaneous Operations Involving Matrices Stack Operation for Matrix Calculations Conditional Tests on Matrix Descriptors Calculating with Matrices Using Matrix Operations in a Program Keystrokes Results Summary of Matrix Functions´m a Calculates residual in result matrix For Further Information Finding the Roots An Equation Using180 Finding the Roots of an Equation Clear program memory ´b0 001-42,21002 003 005 006 007 Finding the Roots of an Equation Desired root Keystrokes ¥ ´ bA000 001-42,21,11 003 004 5000 1 e t Brings another t-valueInto X-register 200 t When No Root Is Found 000 001-42,21 002 003 004 005 Error Choosing Initial Estimates Label 003 004 005 007 X + 8008 009 6 x + 8 Finding the Roots of an Equation Using in a Program Restriction on the Use Memory Requirements Numerical Integration Using f194 002 003 004 4040 1416 7652 Begin subroutine with a label $ ÷ 38254401 6054 Accuracy of f ´ i ´ f 8826 7091 Using f in a Program 382 Memory Requirements Error Conditions Error 0 Improper Mathematics OperationAppendix a 205 Error 1 Improper Matrix Operation Error 2 Improper Statistics Operation Error 3 Improper Register Number or Matrix Element Error 4 Improper Line Number or Label CallError 5 Subroutine Level Too Deep Error 6 Improper Flag Number Pr Error Power Error Stack Lift Last X Register Digit Entry TerminationStack Lift Appendix B Disabling Operations Enabling Operations Stack Stack Enabled. disabled 53.1301 No stack Lift Neutral OperationsAppendix B Stack Lift and the Last X Register Keys Nnn Clear u ¥ Last X Register \ k + H ∆ \ h ÷ À P* q r c ‘ / N z ∕ P\ o j Memory Allocation Memory SpaceAppendix C Registers Appendix C Memory Allocation Memory Status W Memory ReallocationM % Function ´m% 1.0000 Whold 1 64 Restrictions on Reallocation19 ´ m Program Memory Automatic Program Memory Reallocation Memory Requirements for the Advanced Functions Two-Byte Program InstructionsIf executed Together Appendix C Memory Allocation Detailed Look at How WorksAppendix D 220 Appendix D a Detailed Look at Accuracy of the Root X4 = 000 1718 006 007 008 009 010-43,30 011 012-43,30 013 ´ v B Interpreting Results0681 − 45 For 0 x Test for x range Branch for x ≥3x 45x 2 + End subroutine 000.0000 Initial estimates1358 Possible root Appendix D a Detailed Look at ´ b.0 001-42,21,.0 002 003 004 005 Bring x-value into X-register007 008 009 010 013 014 015 016 017 018 10 v ´ ‛ 20 Error 0000 1250 5626 Finding Several Roots Fx = xx a3 = 002 003 004 005 006 007 6667 Same initial estimates Second rootStores root for deflation Deflated function value Deflation for third root Limiting the Estimation Time Counting Iterations For Advanced InformationSpecifying a Tolerance Detailed Look at f How f WorksAppendix E 240 Accuracy, Uncertainty, and Calculation Time X = π1 0π cos4θ − x sinθ dθ 0000 1416 ´ i ´ f Keystrokes Display Return approximation to ´ Clear u HoldKeystrokes ´ i Display ´ f ´ Clear u hold 7858 7807 Uncertainty and the Display Format Functions values for example Δx = 0.5×10−n ×10m = aδx dxb = ab 0.5×10−n + m x dx Conditions That Could Cause Incorrect Results ∞ xe− xdx 001-42,21 002- 1 003 004 005 Appendix E a Detailed Look at f Appendix E a Detailed Look at f Conditions That Prolong Calculation Time Keys lower limit into Keys upper limit intoApproximation to integral Uncertainty Appendix E a Detailed Look at f Obtaining the Current Approximation to an Integral For Advanced Information Low-Power Indication Installing New BatteriesBatteries Batteries Appendix F Batteries Verifying Proper Operation Self-Tests 2.C 3.H Function Summary and Index Complex FunctionsConversions Digit Entry Display Control Index Register ControlLogarithmic Exponential Functions Mantissa. Pressing Matrix Functions Mathematics146 To ZP page164 Number AlterationTo XT Percentage ProbabilityStack Manipulation Clear u Statistics Storage Trigonometry Programming Summary and Index 269 Programming Summary and Index Subject Index 271 Subject Index Subject Index Subject Index Subject Index Subject Index Subject Index Subject Index Subject Index Subject Index Subject Index Subject Index Subject Index Federal Communications Commission Notice Product Regulatory Environment InformationModifications Canadian Notice Avis Canadien European Union Regulatory Notice Body number is inserted between CE