Topic 8 Integration. Exercise 8.2 The fundamental theorem of integral calculus. TOPIC 8 Integration EXERCISE

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1 TOPIC Intgration EXERCISE. Topic Intgration Ercis. Th fundamntal thorm of intgral calculus a Lft nd-point rctangl rul: f., f, f.,. f Approimat ara units b Right nd-point rctangl rul: f.. Approimat ara units f. +. f. +.6 f. +. f Approimat ara units a b cos + sin [ sin cos ] sin cos sin cos Maths Qust Mathmatical Mthods VCE Units and Solutions Manual

2 TOPIC Intgration EXERCISE. a b m 7and n a m m 7 b m m c 7 [ ] m + m + m 7[ ] d m + n m + n 7 + [ ] Maths Qust Mathmatical Mthods VCE Units and Solutions Manual

3 TOPIC Intgration EXERCISE Vrification: ± as a Approimat ara units b Approimat ara units Approimat ara units 9 a f Graph intrscts th ais whr y or Thus a. b f f ; f.; f.7; f Lft End-point Rul: Approimat ara. f + f + f + f units Right End-point Rul: Approimat ara. f + f + f + f units cannot b furthrfactorisd Maths Qust Mathmatical Mthods VCE Units and Solutions Manual

4 6 TOPIC Intgration EXERCISE. a b c +, d sin cos cos cos f cos sin sin cos sin cos sin cos Givn that f 7. and g. a f f 7. Maths Qust Mathmatical Mthods VCE Units and Solutions Manual

5 TOPIC Intgration EXERCISE. 7 b g g. c + + f f 7. + [ ] d g + f g + f 7.+. g f g f. 7. g + g g. f h h h h h + h 7 h h 7 7 h 9 a a a a a a a a Graph cuts th ais whn f. f + f + Thus is a factor. a + 7+ As 7 + has a discriminant such that thr ar no factors. 7 + soa Maths Qust Mathmatical Mthods VCE Units and Solutions Manual

6 TOPIC Intgration EXERCISE. b units a y sin dy cos + sin 6 a b cos cos y + dy 6 dy sin sin [ sin + cos ] sin + cos sin + cos + + b Solv using CAS:,, > + + Solvd using CAS.96 Maths Qust Mathmatical Mthods VCE Units and Solutions Manual

7 TOPIC Intgration EXERCISE. 9 Ercis. Aras undr curvs Ara is 66 units sin + cos + y, [ ] cos + cos + units y Ara is + bysymmtry 6 units Ara 6 a units y Ara is units y y, y b Ara units 7 a Graph intrscts ais whr y. sin + cos sin cos tan tan suggst.66. Sinc tan is ngativ nd quadrant as Thus m,.6779,. Maths Qust Mathmatical Mthods VCE Units and Solutions Manual

8 6 TOPIC Intgration EXERCISE. b Ara.6779 sin + cos.6779 [ cos + sin ] cos sin.6779 cos + sin units a y dy b + by symmtry units 9 a y f 9 Graph cuts th y ais whr, y 9 9. Graph cuts th ais whr y , dy TP s occur whr. dy 9 dy + + dy Whn, y 9. Whn y ±, ± ± 9 6 y.6, 6 6.6, 6,,,,, 9 f 9 Maths Qust Mathmatical Mthods VCE Units and Solutions Manual

9 TOPIC Intgration EXERCISE. 6 b By symmtry ara is units Rquird ara sin + cos [ ] cos + sin cos + sin cos + sin units a y. + + Graph cuts th ais whr y , +,,,,, Thus a, b, c and d. b Ara is units a y a Whn, y a a a Thus y + 6 b Ara of glass is m c Two windows mtrs. Cost $ $ Maths Qust Mathmatical Mthods VCE Units and Solutions Manual

10 6 TOPIC Intgration EXERCISE. a units b Orang rgion units a sin cos [ ] cos cos + b Shadd rgion is units a cm b Ara of whol motif is cm c.7 m 7. cm Scal factor is a.76 units b Solvd using CAS units + + Ercis. Applications y... y... 6 Whn 6, y, thrfor POI 6, Shadd rgion is units y... y or Whn, y 9 9 Whn, y 9 Graphs intrsct at,9 and,. y, 9, y 9, Ara is units Point of intrsction btwn th graphs: sin cos tan y Basic angl, tan is ngativ in th nd quadrant Maths Qust Mathmatical Mthods VCE Units and Solutions Manual

11 TOPIC Intgration EXERCISE. 6 y y sin,,, y cos Ara is A sin + cos + cos sin cos sin [ + ] + [ sin + cos ] cos sin cos sin sin cos sin cos units Ara is units Avrag valu b a b a f.9 b 6 Avrag valu f b a a Maths Qust Mathmatical Mthods VCE Units and Solutions Manual

12 6 TOPIC Intgration EXERCISE. dl 7 t dt t Avrag total incras in lngth is t dt t cm Thrfor, th avrag total incras in lngth is. cm dn a & b.t + dt d Distanc is + t mtrs dt t+ t dn dt 6 dn dt.t +, 6 t a v cos dt t cos dt t 6sin c + Whn t,, c Numbr of brics.t+ dt.t + t brics 9 v + t+ + t + a Initially t, v + m/s dv b a t + dt t + i Whn t, a. m/s ii Whn, + t a 6. m/s c v t,.9 6sin + c 6 + c + c c Thus t 6sin b Whn t, 6sin 6sin 6 6sin 6 m c v t v cos,, 7,, t y + +, t d Distanc is t t cos dt cos dt t cos dt Maths Qust Mathmatical Mthods VCE Units and Solutions Manual

13 TOPIC Intgration EXERCISE. 6 t 6sin 6sin 6sin 6sin 6sin 6+.9 m cos t v dv sin t a dt f Whn t, a sin a sin 6 a sin a m/s a y... y or as + + cannot b furthrfactorisd Whn, y Whn, y Thus ab,, and c,, b Ara is units Maths Qust Mathmatical Mthods VCE Units and Solutions Manual

14 66 TOPIC Intgration EXERCISE. c Avrag valu b a Ara is. cos.. sin. b a f sin.. sin units.. a y... y... Substitut into + or Whn, y Whn, y Points of intrsction ar, and,. b Grn rgion is units c Orang rgion is + + Maths Qust Mathmatical Mthods VCE Units and Solutions Manual

15 units d Ara btwn graphs is grn rgion + orang rgion Ara is a units units y. + y + y, TOPIC Intgration EXERCISE. 67, 6,,, b y y Whn, y + + Whn, y + + Whn, y + Thrfor, th co-ordinats ar,,, and,. Maths Qust Mathmatical Mthods VCE Units and Solutions Manual

16 6 TOPIC Intgration EXERCISE. c Ara btwn curvs is a v units, y. sin +, b Ara is t m.sin + cos + cos + cos c Soil rquird is.. m..t 7 v. dv a a dt.. t.t b. dt.t.t+ c Whn, t. t. + c c.t.t + c Whn t, mtrs Maths Qust Mathmatical Mthods VCE Units and Solutions Manual

17 TOPIC Intgration EXERCISE. 69 d Fourth scond occurs btwn t and t. Distanc.t..t.t dt mtrs a y a + Whn, y a a a a. Thus quation of arch is y.... b. m c Ston ara 7 m. d Volum of stons is m. 7.t 9 N. dt.t million a dc dt 7.7., 7 a b Total cost is t + dt t + t $6,. y y cos y sin, b Ara btwn curvs is cos sin + sin cos units + cos sin Parabola, of th form y a +, a a a Equation of parabola is y + y dc dt t +,, Parabolic canal, t Maths Qust Mathmatical Mthods VCE Units and Solutions Manual

18 7 TOPIC Intgration EXERCISE. If th canal wr full with watr, th cross sctional ara would b 6 m Whn th canal is on-third full, th cross sctional ara m. Cross-sctional ara for a canal which is on third full is givn by A.9.9 y. Thrfor th dpth of watr.. m. Maths Qust Mathmatical Mthods VCE Units and Solutions Manual