APPM 36 Fial Sprig 3. For this proble let the regio R be the regio eclosed by the curve y l( ) ad the lies, y, ad y. (a) (6 pts) Fid the area of the regio R. (b) (6 pts) Suppose the regio R is revolved about the lie y 5. Fid the volue of the resultig solid usig the Shell Method. (c) (6 pts) Suppose we fid a thi piece of etal i the shape of regio R with desity ρ.69 kg/, fid, the -coordiate of the ceter of ass of the thi piece of etal. (d) (6 pts) Suppose the regio R is revolved about the lie. Set up, but do ot solve, a itegral (or itegrals) to fid the volue of the resultig solid usig the Disk/Washer Method. (e) (6 pts) Suppose a surface is geerated by rotatig the curve y l( ), y about the lie y. Set up, but do ot solve, a itegral (or itegrals) to fid the surface area of this surface. (a) A e y dy e y e e e3 e 4.5 - -.75 -.5 -.5. -.8 -.6 -.4-3. -4 (b) Here V πrh y where r y + 5 ad h e y, ow we use itegratio by parts to evaluate the itegral with u y + 5 du dy ad dv e y v e y, thus [ V π(y + 5)e y dy π e y ] (y + 5) e y dy 6πe (c) Here we have ad ote that M y M e y dy ey 4 ey ρey dy ρe y dy e y dy e y dy e 4 + e 8 4 e6 4e 8 ad cobiig this with the result fro part (a), we have e6 e 4 4e 8 e 3 ( + e3 ) 4e 4.
(d) Here V π[r r ] y where R ad r e y ad so V π[() ( e y ) ] dy. (e) Here SA πrdl where r y ad dl + g (y) dy where g(y) e y ad so SA π( y) + g (y) dy π( y) + e y dy πy + e y dy or, with respect to, we have r f() l( ) ad dl + f () d where f() l( ) ad so SA π( f()) e + f () d π( l( )) + e d π l( ) + d e e. The followig probles are ot related: (a) (6 pts) Evaluate the itegral 3 d (b) (6 pts) What error estiate do we have if we approiate the itegral 6 f() d usig a Trapezoidal rule with 8 subitervals of equal legth ad if we kow that 3 f () for all? (c) (6 pts) Deterie if the sequece coverges or diverges: a d, where,, 3,.... (d) (6 pts) Suppose we approiate f() by T (), a Taylor polyoial of order cetered at a, use Taylor s Forula to fid a error boud of the approiatio f() T () if 4 4.. (a) Usig the trigooetric substitutio si(θ), we get 3 d π/ si 3 (θ) cos (θ) dθ ow ote that si 3 (θ) si (θ) si(θ) [ cos (θ)] si(θ) ad so π/ si 3 (θ) cos (θ) dθ π/ [ cos (θ)] cos (θ) si(θ) dθ π/ [cos (θ) cos 4 (θ)] si(θ) dθ ow let u cos(θ) the du si(θ) dθ ad we get π/ [ ] u [cos (θ) cos 4 (θ)] si(θ) dθ (u u 4 3 ) du 3 u5 5 3 5 5 (b) Note that here b a 6 4, 8, ad f () 3 K, so E T (c) Note that d l thus the sequece coverges to. l(), ad so li a l() L li H K(b a)3 3 43 8 4. (d) Note that the error ter here is R () f (z)! ( ) for soe z betwee ad a, also ote that f () 4 3/, ow sice z is betwee ad a ad sice 4 4. we have that < z < 4 ad
so z < ( ) 3/ ( ) 3/ ad thus z 3/ < ad also ote. thus < (.) z 8 ad so the resultig error boud accordig to Taylor s Forula is error R () f (z) ( )!! 4 ( ) 3/ < z! 4 (.) (.) 8 8 8 (.) 4 3. (a) Deterie if the give series are absolutely coverget, coditioally coverget or diverget, fid the su of the series whe possible, justify all aswers: (i) (6 pts.) () (ii) (6 pts.) + + 8 + 4 + 3 + 6 + 6 + (b) Evaluate the itegrals, show all work: (i) (6 pts.) 3 d (ii) (6 pts.) ta () d + (a) (i) Note that () ad ow we perfor a Liit Copariso Test with the diverget haroic series, so we have 6 6 /( ) L li li H / ad sice < <, the series is diverget by the Liit Copariso Test. Now ote that 6 usig the Alteratig Series Test with b, we have that b ( ) ( ) 4 ( ) < for > L ad li H ad so b is decreasig ad goes to ad so () is coditioally coverget by the Alteratig Series Test ad the work doe 6 previously. (a) (ii) Note that, + + 8 + 4 + 3 + 6 + ( ) + + ( ) 4 + ( ) + + ( ) ( ) 4 + 3 4 3/ /4 so this is a coverget Geoetric Series (sice r /4 < ) which coverges absolutely to. (b) (i) Note that 3 + ( + ) ad usig partial fractios we have + ( + ) A + B + C + + (A + B) + C + A A, B, C thus + 3 + ad so + 3 + d d d + l ta () + C + [ d li t + t 3 d li l ta () ] + t + t li t + π 4 [ l t ta (t) ]
+ thus the itegral 3 d diverges to. + (b) (ii) Usig itegratio by parts with u ta () du + ta () d ta () + d }{{} u+ where i the last equality we used the u-substitutio u +. ad dv d v yields ta () l( + ) + C () 4. Suppose you are told that the Maclauri Series of l( + ) is l( + ), ad suppose l( + t) you are give the fuctio F () dt for. t (a) (6 pts) Fid the Maclauri series for the fuctio F (). What is the iterval of covergece of this Maclauri series? (b) (6 pts) Fid the fifth-degree Taylor polyoial, T 5 (), of F () at. (c) (6 pts) Estiate the accuracy of the polyoial i (b) as a approiatio to the defiite itegral F (). (a) F () l( + ) () () li a + a li + li ow ote that + ad so iplies, ow to check the edpoits. If the we have F () () () which is a diverget p-series ad if the we have F () () which is coverget by the Alteratig Series Test (here b / is decreasig ad li b ), so the iterval of covergece is <. (Note the covergece at is coditioal but the proble does ot ask this.) (b) Note that F () l( + t) t dt () t () t () thus F () 4 + 3 9 4 6 + 5 5 6 36 +... ad so T 5() 4 + 3 9 4 6 + 5 5 (c) Fro part (b) ote that F () 4 + 9 6 + }{{ 5 } 36 +... T 5 () ad so by the Alteratig Series Error Estiatio Theore error F () T 5 () 36.
5. The followig probles are ot related: (a) (9 pts.) Sketch the curve 4 si(θ), y cos(θ) for θ π. Be sure to idicate the directio i which the curve is traced (b) (9 pts.) Fid a Cartesia equatio of the paraetric curve cosh(t), y sih(t). Idetify the type of Cartesia curve you have foud. (c) (6 pts.) Fid dy/d of the curve (s) s s, y(s) s 3 whe s. (d) (6 pts) Fid the eact legth of the curve e t + e t, y 5 t, t 3. ( ) (a) Note that si (θ) + cos (θ) iplies + y ad so we have a ellipse, ad ote that whe 4 θ we have (, y) (, ) ad whe θ π we have (, y) (, ), thus,.6..8.4 -.4 - -.6 -. -.8 -.4.4.8..6.4 -.4 -.8 -. -.6 (b) Note that cosh(t) ad sih(t) y/ ad fro the forula sheet we kow cosh (t) sih (t) so we have ( y () ) /4 y 4 or + y 4 which yields a hyperbola (ote techically we oly have the part of the hyperbola above the -ais sice cosh(t) > for all t). (c) Here we have, dy d dy/ds s d/ds 3s 4 s 3 s. (d) Note that (t) e t e t ad y (t) ad so L thus L b a 3 (t) + y (t) dt 3 3 (e t + e t ) dt (e t e t ) 6. The followig probles are ot related: 3 (e t + e t ) dt 3 3 e t + + e t dt (e t + e t ) dt (e t + e t ) dt e 3 e 3 e6 e 3. (a) (9 pts.) Fid the area eclosed by the curve r si(θ). (b) (9 pts.) Fid the area iside r si(θ) ad outside r.
(c) (6 pts.) Fid the area of oe loop of r si(θ). (a) Usig syetry, we have [ ] 3π/ 3π/ A ( si(θ)) dθ ( si(θ) + si (θ)) dθ π/ π/ 3π/ ( si(θ) + ) 3π/ ( cos(θ)) dθ π/ [ ] 3 si(θ) 3π/ θ + cos(θ) 4 π/ π/ 9π 4 3π 4 6π 4 3π ( 3 si(θ) ) cos(θ) (b) Note that these curves itersect whe θ or π ad usig the work doe above, we have [ 3π/ A ( si(θ)) dθ ] 3π/ [ ] 3 () si(θ) 3π/ dθ θ + cos(θ) θ π π 4 π [ ] si(θ) 3π/ θ + cos(θ) 3π ( π ) 4 4 π 4 + π dθ.4.6.8-3. -.4 -.6 -.8.8.6.4 3. 4 -.8 -.6 -.4-3. (c) Usig syetry we have A [ π/4 ( si(θ) ) dθ] π/4 si(θ) dθ cos(θ) π/4