Higher Engineering Mathematics

382 INTEGRAL CALCULUS When areaPQRSis rotated about axisXXthe vol- ume generated is that of the pulley. The centroid of the semi ...

SOME APPLICATIONS OF INTEGRATION 383 H from the centroid of the area to the fixed axis is squared. Second moments of areas are u ...

384 INTEGRAL CALCULUS Figure 38.16 centroid may be determined. In the rectangle shown in Fig. 38.16,Ipp= bl^3 3 (from above). Fr ...

SOME APPLICATIONS OF INTEGRATION 385 H Table 38.1 Summary of standard results of the second moments of areas of regular sections ...

386 INTEGRAL CALCULUS IPP=Ak^2 PP, from which, kPP= √ IPP area = √( 645000 600 ) =32.79 mm Problem 13. Determine the second mome ...

SOME APPLICATIONS OF INTEGRATION 387 H The centroid of a semicircle lies at 4 r 3 π from its diameter. Using the parallel axis t ...

388 INTEGRAL CALCULUS Problem 18. Determine correct to 3 significant figures, the second moment of area about axis XXfor the com ...

SOME APPLICATIONS OF INTEGRATION 389 H For rectangle F: IXX= bl^3 3 = (15.0)(4.0)^3 3 =320 cm^4 Total second moment of area for ...

390 INTEGRAL CALCULUS Figure 38.31 Calculate the radius of gyration of a rectan- gular door 2.0 m high by 1.5 m wide about a ve ...

H Integral calculus 39 Integration using algebraic substitutions 39.1 Introduction Functions which require integrating are not a ...

392 INTEGRAL CALCULUS Letu=(5x−3) then du dx =5 and dx= du 5 Hence ∫ 4 (5x−3) dx= ∫ 4 u du 5 = 4 5 ∫ 1 u du = 4 5 lnu+c= 4 5 ln( ...

INTEGRATION USING ALGEBRAIC SUBSTITUTIONS 393 H 7. ∫ 1 0 (3x+1)^5 dx [227.5] 8. ∫ 2 0 x √ (2x^2 +1) dx [4.333] 9. ∫ π 3 0 2 sin ...

394 INTEGRAL CALCULUS It is possible in this case to change the limits of inte- gration. Thus whenx=3,u=2(3)^2 + 7 =25 and whenx ...

INTEGRATION USING ALGEBRAIC SUBSTITUTIONS 395 H The electrostatic potential on all parts of a conducting circular disc of radiu ...

Assign-10-H8152.tex 23/6/2006 15: 11 Page 396 Integral calculus Assignment 10 This assignment covers the material contained in C ...

H Integral calculus 40 Integration using trigonometric and hyperbolic substitutions 40.1 Introduction Table 40.1 gives a summary ...

398 INTEGRAL CALCULUS Table 40.1 Integrals using trigonometric and hyperbolic substitutions f(x) ∫ f(x)dx Method See problem co ...

INTEGRATION USING TRIGONOMETRIC AND HYPERBOLIC SUBSTITUTIONS 399 H Now try the following exercise. Exercise 156 Further problems ...

400 INTEGRAL CALCULUS = [ 3 θ 2 +sin 2θ+ sin 4θ 8 ]π 4 0 = [ 3 2 (π 4 ) +sin 2 π 4 + sin 4(π/4) 8 ] −[0] = 3 π 8 + 1 = 2. 178 , ...

INTEGRATION USING TRIGONOMETRIC AND HYPERBOLIC SUBSTITUTIONS 401 H Problem 11. Evaluate ∫ 1 0 2 cos 6θcosθdθ, correct to 4 decim ...