Higher Engineering Mathematics

362 DIFFERENTIAL CALCULUS Hence the stationary points are (0, 0) and (2, 0). (iv) ∂^2 z ∂x^2 = 6 x−6, ∂^2 z ∂y^2 =−8 and ∂^2 z ∂ ...

MAXIMA, MINIMA AND SADDLE POINTS FOR FUNCTIONS OF TWO VARIABLES 363 G z x y Figure 36.11 Substituting in equation (2) gives: S=x ...

364 DIFFERENTIAL CALCULUS Locate the stationary points on the surface f(x,y)= 2 x^3 + 2 y^3 − 6 x− 24 y+ 16 and determine thei ...

Assign-09-H8152.tex 23/6/2006 15: 11 Page 365 G Differential calculus Assignment 9 This assignment covers the material contained ...

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Integral calculus H 37 Standard integration 37.1 The process of integration The process of integration reverses the process of d ...

368 INTEGRAL CALCULUS Table 37.1 Standard integrals (i) ∫ axndx= axn+^1 n+ 1 +c (except whenn=−1) (ii) ∫ cosaxdx= 1 a sinax+c (i ...

STANDARD INTEGRATION 369 H Problem 4. Determine ∫ 3 x^2 dx. ∫ 3 x^2 dx= ∫ 3 x−^2 dx. Using the standard integral, ∫ axndxwhena=3 ...

370 INTEGRAL CALCULUS (a) From Table 37.1(iv), ∫ 7 sec^24 tdt=(7) ( 1 4 ) tan 4t+c = 7 4 tan 4t+c (b) From Table 37.1(v), 3 ∫ co ...

STANDARD INTEGRATION 371 H (a) ∫ 3 4 sec^23 xdx (b) ∫ 2 cosec^24 θdθ [ (a) 1 4 tan 3x+c(b)− 1 2 cot 4θ+c ] (a) 5 ∫ cot 2tcos ...

372 INTEGRAL CALCULUS = ⎡ ⎣θ 3 2 3 2 + 2 θ 1 2 1 2 ⎤ ⎦ 4 1 = [ 2 3 √ θ^3 + 4 √ θ ] 4 1 = { 2 3 √ (4)^3 + 4 √ 4 } − { 2 3 √ (1)^3 ...

STANDARD INTEGRATION 373 H (a) ∫ 2 1 cosec^24 tdt (b) ∫π 2 π 4 (3 sin 2x−2 cos 3x)dx [(a) 0.2527 (b) 2.638] (a) ∫ 1 0 3e^3 t ...

Integral calculus 38 Some applications of integration 38.1 Introduction There are a number of applications of integral calcu- lu ...

SOME APPLICATIONS OF INTEGRATION 375 H By firstly determining the points of intersection the range ofx-values has been found. Ta ...

376 INTEGRAL CALCULUS 38.3 Mean and r.m.s. values With reference to Fig. 38.5, mean value,y= 1 b−a ∫b a ydx and r.m.s. value= √ ...

SOME APPLICATIONS OF INTEGRATION 377 H Now try the following exercise. Exercise 149 Further problems on mean and r.m.s. values ...

378 INTEGRAL CALCULUS At the points of intersection the co-ordinates of the curves are equal. Sincey=x^2 theny^2 =x^4. Hence equ ...

SOME APPLICATIONS OF INTEGRATION 379 H where it balances perfectly, i.e. the lamina’scen- tre of mass. When dealing with an area ...

380 INTEGRAL CALCULUS = 625 3 − 625 4 125 2 − 125 3 = 625 12 125 6 = ( 625 12 )( 6 125 ) = 5 2 =2.5 y= 1 2 ∫ 5 0 y^2 dx ∫ 5 0 yd ...

SOME APPLICATIONS OF INTEGRATION 381 H Figure 38.12 (b) (i) When the shaded area of Fig. 38.12 is revolved 360◦about thex-axis, ...