Higher Engineering Mathematics, Sixth Edition
202 Higher Engineering Mathematics Graphical solutions to Problems 6 and 7, Exercise 80, page 195 26 24 24 22 22 6 4 2 0 2 4 6 y ...
Chapter 19 Irregular areas, volumes and mean values of waveforms 19.1 Areas of irregular figures Areas of irregular plane surfac ...
204 Higher Engineering Mathematics (iii) Accurately measure ordinatesy 1 ,y 2 ,y 3 ,etc. (iv) AreaABCD=d(y 1 +y 2 +y 3 +y 4 +y 5 ...
Irregular areas, volumes and mean values of waveforms 205 Problem 2. A river is 15m wide. Soundings of the depth are made at equ ...
206 Higher Engineering Mathematics A sketch of the tree trunk is similar to that shown in Fig. 19.5 above, where d=2m, A 1 = 0 . ...
Irregular areas, volumes and mean values of waveforms 207 dd b y 1 y 2 y 3 y 4 y 5 y 6 y 7 y dddd d Figure 19.6 If the mid-ordin ...
208 Higher Engineering Mathematics (b) Area under waveform (b) for a half cycle=( 1 × 1 )+( 3 × 2 )=7As. Average value of wavefo ...
Irregular areas, volumes and mean values of waveforms 209 50 Graph of power/time 40 30 Power (kW) 20 10 0 123456 Time (hours) 7. ...
210 Higher Engineering Mathematics (a) The width of each interval is 12. 0 6 cm. Using Simpson’s rule, area=^13 ( 2. 0 )[( 3. 6 ...
Irregular areas, volumes and mean values of waveforms 211 An indicator diagram of a steam engine is 12cm long. Seven evenly spa ...
Revision Test 6 This Revision Test covers the material contained in Chapters 18 and 19.The marks for each question are shown in ...
Chapter 20 Complex numbers 20.1 Cartesian complex numbers There are several applications of complex numbers in science and engin ...
214 Higher Engineering Mathematics Problem 3. Evaluate (a)j^3 (b)j^4 (c) j^23 (d) − 4 j^9 (a) j^3 =j^2 ×j=(− 1 )×j=−j,sincej^2 = ...
Complex numbers 215 Thus, for example, ( 2 +j 3 )+( 3 −j 4 )= 2 +j 3 + 3 −j 4 = 5 −j 1 and ( 2 +j 3 )−( 3 −j 4 )= 2 +j 3 − 3 +j ...
216 Higher Engineering Mathematics 20.4 Multiplication and division of complex numbers (i) Multiplicationof complex numbersis ac ...
Complex numbers 217 (a) ( 1 +j)^2 =( 1 +j)( 1 +j)= 1 +j+j+j^2 = 1 +j+j− 1 =j 2 ( 1 +j)^4 =[( 1 +j)^2 ]^2 =(j 2 )^2 =j^24 =− 4 He ...
218 Higher Engineering Mathematics (b) ( 1 +j 2 )(− 2 −j 3 )=a+jb − 2 −j 3 −j 4 −j^26 =a+jb Hence 4−j 7 =a+jb Equating real and ...
Complex numbers 219 (iii) θis called theargument(or amplitude) ofZand is written as argZ. By trigonometry on triangleOAZ, argZ=θ ...
220 Higher Engineering Mathematics (a) (b) Real axis 4 0 308 Imaginary axis x jy Real axis 7 1458 x jy Figure 20.7 Using trigo ...
Complex numbers 221 =( 1. 732 +j 1. 000 )+( 3. 536 −j 3. 536 ) −(− 2. 000 +j 3. 464 ) = 7. 268 −j 6. 000 ,which lies in the four ...
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