Question

#43 and #47 please

Surface Area - Surfaces of Revolution In Exercises 43–70, a curve C with parametrization = X(t), y=0, z = 2(t), for t in (a, b), is revolved about the z-axis to create a surface S. Find the surface area A of S. NOTE: Only the first four prob- lems are like Example 13.2.7 (a) and you will have to use integrals over two subintervals of [a, b] to compute A. 13. X = t3 – įt, Z = 2t?, [0, 1] 44. X = 2t - +3, 2 = 3t², (0,2] 45. X = t - fts, Z = 2+, [0, 1] 46. X = 2 – 346, Z = t4, [0, il 47.) X = 2t3, Z = 9t - , [0,1]

Answer 1

3 dx de X=tgt Z=2ť 32-1 / 3 2 2 2 2 = 46 The sirface generated by the revolection te about zaxis sirface Then A= 28 2t V(3+= 1 2 3 4 (46) It Il s the area be 0 € Formula n=f.ft) y=9(+) Asier Stay Valde Now A = 20 do 27 V 8144 2 € ✓ 8.144 18++ 1 + 16xgt at = 21 S 27 V 81+4+14X8*+1 dt 2. 2 / 2 / 31 22 14 x 92 H d 2 24 3 2,11 3 ť v81444 9114 2+1 Sol 410 So 't VT #26 € + 81 24 unf الى 3 411 x 3.3685 3 Square to
1
( 47 6t2 X=2t3 = gt-1/t5 dx dt diz at g- 444 - А. zob.es 9 dt 1 21 25 4 th ) / (66%) 788-919 So' @S-4+9)/2804°3*4(95–929504 2.7 x 1957.77 square ienet Here A be the area of revolved about z axis create a Sirface where x = '2t? yzo Z.=9-444 25 (+) si
2
integrate 0 to 1 (45-4t-4)(30t^4+(45-4t^, Σο Examples Assuming "O to 1" is referring to variable ranges Definite integral: S" (45 – 4+4) / 30+* +(45 – 44*)? dt = 1957.77
suported 1
Wolfram Alpha computational intelligence itegrate 0 to 11t 2(81t^4+9x14t^2+1)^1/ 6 SES Examples Assuming "0 to 1" is referring to variable ranges Definite integral: $*2 V 1+126 tº +81 +* dt = 3.3685
suported 2
J (45-4 (45 – 4t), 30 tº + (45 – 4t*) dt = t-3546 - 16 it4 + 15 23 +165 i 16 it4 15+ 23 +11 i 23 +-11 i (5 1 1 9 16 it4 16t4 F 4' 2' 2' 4' 15 (11i+V23) 15 (11 +i V23) - 16 it4 + 15 23 + 165 i 16 itº 196 425 15+ 23 +11 i 23 +-11 (11 15 16 itt 16 it4 F1 4' 2' 2' 4' 15 (11i + V23) 165 i - 15 /23 320 +12 + 15 720 +8 - 228 600 ++ +1 154 250 154250) (45 V 16t® - 330 +4 + 2025 ) + constant -
3
3t Se81t* +9_14t? +1 dt = 8 i (193 V 3 – 336) V9+2 +4 73 +7 79(7+473)+2 + 1 97 +56 V3 1/3) 386 i(4 73 -7) / 942 +4 73 +7 /9(7+4/3) ++ +1 97 +56 V3+ F|i sinh 7+43 3t (2187t“ + 4536 +4 + 1791t+ 14) (405 V 81t* +126² +1 ) +constant i
4