Hp 33s User Manual Page 344
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E–6 More about Integration
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)
x
Calculated integral
of this function
will be accurate.
f (x)
a b
x
Calculated integral
of this function
may be inaccurate.
a b
- Edition 3 1
- HP part number F2216-90001 1
- Printing History 2
- Contents 3
- 2 Contents 4
- 4. Real–Number Functions 5
- 5. Fractions 6
- 7. Solving Equations 7
- 8. Integrating Equations 7
- 13. Programming Techniques 10
- Appendixes and Reference 11
- B. User Memory and the Stack 11
- C. ALG: Summary 12
- D. More about Solving 12
- E. More about Integration 13
- F. Messages 13
- G. Operation Index 13
- Basic Operation 15
- Getting Started 17
- Shifted Keys 18
- Alpha Keys 19
- Cursor Keys 19
- Silver Paint Keys 20
- Backspacing and Clearing 20
- Getting Started 1–5 21
- Keys for Clearing 21
- Key Description 21
- 1–6 Getting Started 22
- Using Menus 23
- 1–8 Getting Started 24
- Exiting Menus 25
- RPN and ALG Keys 26
- First Line 27
- Second Line 27
- Annunciators 27
- Getting Started 28
- HP 33s Annunciators 28
- Getting Started 1–13 29
- Keying in Numbers 30
- Understanding Digit Entry 31
- Doing Arithmetic 32
- One–Number Functions 33
- Two–Number Functions 33
- Number of Decimal Places 35
- Fractions 37
- Calculator Memory 40
- RPN: The Automatic 41
- Memory Stack 41
- Clearing the X–Register 42
- How ENTER Works 45
- How CLEAR x Works 46
- 3947.52704.96 + 49
- Exercises 53
- Order of Calculation 53
- More Exercises 54
- 5961.0)7.05.12()8.04.5( 55
- Storing Data into Variables 57
- Clearing Variables 60
- Recall Arithmetic 61
- Real–Number Functions 65
- Entering π 67
- Setting the Angular Mode 68
- Trigonometric Functions 68
- Real–Number Functions 4–5 69
- Hyperbolic Functions 70
- Percentage Functions 70
- Real–Number Functions 4–7 71
- 4–8 Real–Number Functions 72
- Real–Number Functions 4–9 73
- Items Description Value 73
- Real–Number Functions 4–11 75
- Time Conversions 76
- Angle Conversions 77
- Unit Conversions 77
- Probability Functions 78
- Real–Number Functions 4–15 79
- Parts of Numbers 80
- Names of Functions 81
- Fractions in the Display 84
- Accuracy Indicators 85
- Longer Fractions 86
- Choosing a Fraction Format 87
- 5–6 Fractions 88
- Fractions 5–7 89
- Fractions 90
- Fractions in Programs 91
- How You Can Use Equations 93
- Key Operation 95
- Variables in Equations 96
- Numbers in Equations 97
- Functions in Equations 97
- Parentheses in Equations 98
- Using ENTER for Evaluation 103
- Using XEQ for Evaluation 104
- The Syntax of Equations 105
- Order Operation Example 106
- Equations Meaning 106
- P=A+B+Hx(1 SIN(T)+1 SIN(F)) 109
- Verifying Equations 110
- Solving Equations 111
- Solving Equations 112
- Solving Equations 7–3 113
- Solving Equations 7–5 115
- Verifying the Result 116
- Solving Equations 7–7 117
- 7–8 Solving Equations 118
- Solving Equations 7–9 119
- 7–10 Solving Equations 120
- For More Information 121
- Integrating Equations 123
- Integrating Equations ( ³ FN) 124
- Integrating Equations 8–3 125
- Integrating Equations 126
- Accuracy of Integration 127
- Specifying Accuracy 128
- Interpreting Accuracy 128
- Integrating Equations 8–7 129
- Operations with 131
- Complex Numbers 131
- Complex Operations 132
- BASE Menu 139
- Menu label Description 139
- Negative Numbers 142
- Range of Numbers 143
- Press to display 144
- left window 144
- right window 144
- Statistical Operations 145
- Entering One–Variable Data 146
- Entering Two–Variable Data 146
- Statistical Operations 11–3 147
- Initial x, y Corrected x, y 147
- Statistical Operations 148
- Statistics Menus 148
- Menu Key Description 148
- Statistical Operations 11–5 149
- Sample Standard Deviation 150
- Linear Regression 151
- Statistical Operations 11–9 153
- Summation Statistics 154
- Statistical Operations 11–11 155
- 11–12 Statistical Operations 156
- Programming 157
- Simple Programming 159
- Designing a Program 161
- Data Input and Output 162
- Entering a Program 163
- Keys That Clear 164
- Function Names in Programs 165
- Simple Programming 166
- (In RPN mode) 166
- Display: Description: 166
- Running a Program 167
- Entering and Displaying Data 169
- 12–12 Simple Programming 170
- RPN mode ALG mode 170
- Simple Programming 12–13 171
- Simple Programming 12–15 173
- Error Stops 175
- Editing a Program 176
- Program Memory 177
- Memory Usage 178
- The Checksum 179
- Nonprogrammable Functions 180
- Programming with BASE 180
- Simple Programming 12–23 181
- Simple Programming 12–25 183
- Programming Techniques 185
- 13–2 Programming Techniques 186
- MAIN program 187
- (top level) 187
- End of program 187
- DCBA +++ 188
- Conditional Instructions 190
- Keys: Display: 191
- Programming Techniques 13–7 191
- The Test Menus 191
- 13–8 Programming Techniques 192
- Programming Techniques 13–9 193
- Fraction–Control Flags Flag 193
- 7 8 9 193
- Programming Techniques 194
- Programming Techniques 13–11 195
- FLAGS Menu 195
- Menu Key Description 195
- 13–12 Programming Techniques 196
- Programming Techniques 13–13 197
- Program Lines: 197
- Description: 197
- Programming Techniques 13–15 199
- (In ALG mode) 199
- Conditional Loops (GTO) 201
- Programming Techniques 13–19 203
- The Variable "i" 204
- The Indirect Address, (i) 205
- Program Control with (i) 206
- Programming Techniques 13–23 207
- Program lines: 207
- Equations with (i) 208
- Programming Techniques 13–25 209
- Solving a Program 211
- Using SOLVE in a Program 216
- Integrating a Program 217
- Mathematics Programs 223
- 15–2 Mathematics Programs 224
- ++ and P 225
- 15–4 Mathematics Programs 226
- Mathematics Programs 15–5 227
- Description 227
- 15–6 Mathematics Programs 228
- Mathematics Programs 15–7 229
- Mathematics Programs 230
- Mathematics Programs 15–9 231
- Mathematics Programs 15–13 235
- 15–14 Mathematics Programs 236
- Mathematics Programs 15–15 237
- 15–16 Mathematics Programs 238
- Mathematics Programs 15–17 239
- Mathematics Programs 15–19 241
- Mathematics Programs 15–21 243
- 15–22 Mathematics Programs 244
- Mathematics Programs 15–23 245
- 15–24 Mathematics Programs 246
- Mathematics Programs 15–25 247
- 15–26 Mathematics Programs 248
- Mathematics Programs 15–27 249
- 15–28 Mathematics Programs 250
- Mathematics Programs 15–29 251
- Terms and Coefficients 251
- Mathematics Programs 15–31 253
- Coordinate Transformations 254
- Mathematics Programs 15–33 255
- 15–34 Mathematics Programs 256
- Mathematics Programs 15–35 257
- Mathematics Programs 15–37 259
- Statistics Programs 261
- 16–2 Statistics Programs 262
- Statistics Programs 16–3 263
- 16–4 Statistics Programs 264
- Statistics Programs 16–5 265
- (Y – B) ÷ M 266
- Statistics Programs 16–7 267
- Statistics Programs 268
- Statistics Programs 16–11 271
- 16–12 Statistics Programs 272
- ÷÷−− SMX 273
- Statistics Programs 16–15 275
- Statistics Programs 16–17 277
- 16–18 Statistics Programs 278
- Statistics Programs 16–19 279
- Statistics Programs 16–21 281
- Miscellaneous Programs and 283
- Equations 283
- Prime Number Generator 288
- Support, Batteries 295
- Environmental Limits 296
- Changing the Batteries 296
- Testing Calculator Operation 298
- The Self–Test 299
- Warranty 300
- Service 301
- Regulatory Information 303
- User Memory and the Stack 305
- Resetting the Calculator 306
- Clearing Memory 307
- The Status of Stack Lift 308
- User Memory and the Stack 310
- ALG: Summary 311
- Simple Arithmetic 312
- Power Functions 312
- Percentage Calculations 313
- ALG: Summary 314
- Parentheses Calculations 315
- Chain Calculations 315
- Reviewing the Stack 316
- Coordinate Conversions 317
- Integrating an Equation 318
- ALG: Summary C–9 319
- ALG: Summary C–13 323
- More about Solving 325
- D–2 More about Solving 326
- Interpreting Results 327
- More about Solving 328
- More about Solving D–5 329
- More about Solving D–7 331
- D–8 More about Solving 332
- More about Solving D–9 333
- 05.00.3)](x[x =−+÷ 335
- 010-]101)x[( 337
- Underflow 338
- More about Integration 339
- E–2 More about Integration 340
- More about Integration E–3 341
- More about Integration 342
- More about Integration E–5 343
- E–6 More about Integration 344
- More about Integration E–7 345
- E–8 More about Integration 346
- More about Integration E–9 347
- Messages 349
- Messages 350
- Messages F–3 351
- Operation Index 353
- Operation Index G–3 355
- Operation Index G–5 357
- Operation Index G–7 359
- Operation Index G–9 361
- Operation Index G–11 363
- Operation Index G–13 365
- Operation Index G–15 367
- Operation Index G–17 369
- Operation Index G–19 371
- G–20 Operation Index 372
- Special Characters 373
- BIN annunciator, 10–1 374
- Index–3 375
- EQN annunciator 376
- EQN annunciator, 6–4 376
- Index–5 377
- HEX annunciator, 10–1 378
- Index–7 379
- Index–9 381
- Index–10 382
- Index–11 383
- Index–13 385
- Index–14 386
- Index–15 387
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