Question

Problem 3. Consider the curve {> 1, y = 1/x}. Compute the length of the part of this curve lying to the left of the line x = a for any a > 1. Show that the length of the whole curve is thus infinite. Compute the area of the surface obtained by rotating this curve about about the x axis by computing the corresponding improper integral; it should be infinite. What is the area of the surface lying to the left of the plane x = a for a > 1? Finally, compute the volume of the surface of revolution obtained by rotating the area below the above curve (ie the region {er > 1,0 < < 1/}) around the 3 axis; it should be finite. (Be careful with the improper integrals inherent in the problem above!)

problem 3 pls

Answer 1

ANSWER:-

9 consider the curve. {x1, y = 12} area of part of the left of line aaa, where as ya (016) X=1 x=a a nal and Area bounded left to a=a area: gay dx = sa l xax = 1 enn) Area = ena = lno-ena. .Area=lna ③ area of whole curve COO) 241

Area = you s lada - [enn] Area = lim ena-lnialno Thus area of whole curve is "infinite area of sustace by rotating about a-auis. ② 1016) where 12ALOO surface area salanyas. ds: It (dy ldx)2 da surface areals) > 2015 yos. S = 27 ly it (ayldaj? da yox o dy idx=-1/22 S = 275 land it'llay dx 5=2015 JANEIX

put 94 +1=1? > 493 d = 2tdt 22302 - tat: 2015 t (dt) =?"Strat S2 x3 (223) 727 23 S2 . Ś2 t2dt ct2_1312 Surface Areals) - Sa t2dt - Tt²-1) 3/2 1 S2 (t²-1)+1 dt (t²-1) 3/2 eln (0) since i dt . . Szt2-1 Therefore, i dt 12 lt2-1) 3/2 by any value. surface areals). Teads to ao landinity) Area of surface lying to the rest of a-a ☺ where a>

x=1 uza surface area is) a 25 19 yit (dy / 2² an sa 275 X J otllands 3 =21 1x4+1 : 22.da where al put. 2"+1=42 ay 2nd a = tat Jauti , sauti S = 20 tltdt) - zat Se 7 (t²-1)3/2 52 (+²1) 3/2 put to seco, dt Seco ano do secu a4+1) seco KAOtang do 14 (sec²0-i) 3/2 (Tauti) sedotano do ctano)

sed crauti) J recodo oly - tano STA a see Crauti (cos?oj do = sieranti 1 do Tily coso. sino . Colso. Sin? O Sauti so; surface area- ง8 0314 coso. sino volum volunt. I ag ana ing 12 dx = (-1/2) volument (10.1) volumes - 1 volume (V) = π