Chapter 9. Arc Length and Surface Area

Transcription

1 Chpter 9. Arc Length nd Surfce Are In which We ppl integrtion to stud the lengths of curves nd the re of surfces. 9. Arc Length (Tet ) P n P 2 P P 2 n b P i ( i, f( i )) P i ( i, f( i )) distnce from P i to P i ( i i ) 2 + ( f( i ) f( i ) ) 2

2 28 Chpter 9. Arc Length nd Surfce Are let i i be the sme for ll i n s P i P i i n i ( ) 2 + ( f( i ) f( i ) ) 2 n ( ) 2 f(i ) f( i ) + i n + f (c i ) 2 i for some i < c i < i b the M. V. Th. tke the limit s s b + ( f () ) 2 d notice tht for the limit (integrl) to eist we need some hpothesis on the integrnd continuit of f will do Emple 9.. Find the circumference of circle. 2 2

3 d d Arc Length (Tet ) 29 s ( 2 2 ) 2 d lim R [Arcsin ] R d d [Arcsin() Arcsin()] 4 π 2 2π Emple 9..2 Find the length long the curve from to 9. L 9 + d d 2 ( ) 2 d + d d this is not n es integrl d substitute t 2 4; 2t 4 d t L 6 t2 + t t t2 + now the substitution is t tn θ the integrl tht results is sec 3 θ dθ

4 L i 3 Chpter 9. Arc Length nd Surfce Are lternte method interchnge the roles of nd Find the length of 2 from to 3 L d d ( ) 2 d 3 d d + 42 d substitute 2 tn θ to give sec 3 θ dθ Emple 9..3 Find the length long the curve ln from to e. L d d e e + ( ) 2 d d d + e 2 d 2 + d substitute t to give t 2 t Arc Length of Prmetric Curves (Tet ) pproimte length b stright lines L ( i ) 2 + ( i ) 2 i i f(t i ) i f (c i ) t b MVTh. i g(t i ) i g (d i ) t b MVTh. (f (c i ) 2 + (g (d i ) 2 t

5 9.2 Arc Length of Prmetric Curves (Tet ) 3 L β α (d ) 2 + ( ) 2 d requires f (t), g (t) continuous for α t β the curve is trversed once gives old formul if prmeter is Emple 9.2. The circumference of circle. prmetric equtions of circle cos t, sin t; t 2π L 2π 2π 2π (d ) 2 ( ) 2 d + ( sin t)2 + ( cos t) 2 t 2π 2π Emple Find the length of the curve t 2 t; t 2 + t from t /2 to t 3/2. L 3/2 /2 (2t )2 + (2t + ) 2 3/2 8t /2 2 (2t)2 + /2 /2 to do this integrl substitute tn θ 2t recll sec θ ln sec θ + tn θ

6 32 Chpter 9. Arc Length nd Surfce Are Emple Find the length of the curve 4 cos 3 t; 4 sin 3 t. L 2π [2 cos 2 t( sin t)] 2 + [2 sin 2 t(cos t)] 2 2 2π cos t sin t ( sin 2 ) t π/ ! Wht hppens if we ignore the bsolute vlue signs? 9.3 Arc Length of Polr Curves (Tet ) length of sector of circle r r s compre length long the curve with length long the rc of circle ( s) 2 ( r) 2 + (r θ) 2 ( ( ) ) 2 r + r 2 ( θ) 2 θ s β α ( ) 2 dr + r dθ 2 dθ β α (f (θ) ) 2 + ( f(θ) ) 2 dθ

7 9.3 Arc Length of Polr Curves (Tet ) 33 formul cn lso be obtined from using ( ) 2 d + dθ r cos θ r sin θ ( ) 2 d dθ dθ Emple 9.3. Find the circumference of circle. circle, centre (, ), rdius is r circumference C 2π dθ 2π θ 2π dθ 2π Emple r sin 2θ. Find the rclength of one lef of review sketch (emple 8.7.4) one lef is swept out when θ π 2 s π 2 (2 cos 2θ)2 + (sin 2θ) 2 dθ this integrl is not es to do

8 34 Chpter 9. Arc Length nd Surfce Are 9.4 Surfce Are (Tet ) b pproimte the surfce using clinders rdius of the clinder is f(c i ) height of clinder is length long the curve for, length is + ( f (c i ) ) 2 s S 2πf(c i ) + ( f (c i ) ) 2 2πf(ci ) s S 2π b f() + ( f () ) b 2 d 2π f() ds

9 9.4 Surfce Are (Tet ) 35 rottion bout the -is rdius of the clinder is length long the curve s bove S 2π b + ( f () ) b 2 d 2π ds Emple 9.4. Find the surfce re of sphere. 2 2 S 2π b 2 2π 4π + ( ) 2 d ( ) d 2 2 ( ) 2 2 d 2 2 4π d 4π 2

10 36 Chpter 9. Arc Length nd Surfce Are Emple Find the surfce re of cone. r 3'-6" h 3'-6" h r S 2π 2π r + + ( ) 2 h d r ( ) 2 [ h 2 r 2 ] r ( ) 2 [ ] h r 2 2π + r 2 πr r 2 + h 2 Emple Find the surfce re of the object formed when 3/2 from to 9 is revolved bout the -is. S 2π 9 + ( ( ( 3/2 ) ) 2 d 2π ) ds 2π 2π ( ) /2 d d

11 9.4 Surfce Are (Tet ) 37 b first formul with S 2π 2/3 3/ ( ) 2 d d d ( 2π ) ds 2π 27 ( ) 2 2 2/3 + 3 /3 d Emple Find the volume nd surfce re of the infinitel long horn formed when is rotted bout the -is from to. V π 2 d π ( ) 2 R d d π lim R 2 [ ] R [ ] π lim π lim R R R + π( + ) π Conclusion: The horn cn be filled with π units of pint.

12 38 Chpter 9. Arc Length nd Surfce Are S 2π 2π + ( ) 2 d + ( ) 2 2 d 4 + 2π 4 d this integrl need not be evluted notice for R d d lim R lim R ln R lim R ln R b comprison d Conclusion: The surfce cnnot be pinted! 9.5 Surfce Are nd Prmetric Curves ( ) S 2π ds for rottion bout the -is for prmetric f(t), g(t), just showed (d ) 2 ds + needs f, g continuous nd g ( ) 2 d

13 9.5 Surfce Are nd Prmetric Curves ( ) 39 summr for rottion bout the -is S β α (d ) 2 2π + ( ) 2 d for rottion bout the -is S δ γ (d ) 2 2π + ( ) 2 d Emple 9.5. The surfce re of sphere. prmetric equtions of semicircle cos t, sin t; t π rotte bout the -is to get sphere ds for the circle S π 2π ds 2π π 2π 2 ( cos t) π 4π 2 ( sin t) Emple The surfce re of e t t, 4e t/2, t, revolved bout the -is. S 2π 4e t/2 (e t ) 2 + (2e t/2 ) 2 8π e t/2 e 2t 2e t + + 4e t 8π e t/2 (e t + ) 2 8π e 3t/2 + e t/2...

14 4 Chpter 9. Arc Length nd Surfce Are Emple The surfce re of e t t, 4e t/2, t, revolved bout the -is. S 2π (e t t) (e t ) 2 + (2e t/2 ) 2 2π (e t t) e 2t 2e t + + 4e t 2π (e t t)(e t + ) 2π (e 2t te t + e t t)...