Question

E MATH213_Lecture20.pdf Home Tools MATH213 Lectur... x MATH213 Lectur... Class Roster_MAT... Sian In Talking A 1 / 12 142% je & 12 + DIGI Sample Problems Problem 1- Let G be the solid in 3-space defined by the inequalities 1-e" <y <3 – e", 1-y < 22 < 2 – y, y se" <y+4. Compute the volume of G be using the coordinate transformation u = e" + y, v= y + 22, w=e" – y. 1

Answer 1

Problem 1: solution. we have to find the volume that- such of the solid s 1-ex LY L 3-et using the u= ex + y 1-Y 2 22 22-4, y Lex L y +4 coordinate trans for mation- v=y+az ,W=ety that the volume of the solid we Know s ust v=SSS dx dy dz S the Now, if we change then S' Co-ordinate anip the solid in ure, w space un whose volume Same as of s Va SSS todudu do dudvdc. ['Jl ip Jacoblan and S' let Now Giver: find the limits of integration 1-ex L y L 3-ex 1 L ex ty < 3 - 12 u 23 1-4 <2Z LQ-y y Lex Lyt4 = oL ex y z 4 oL L w 24 12 y +22 22 [1 2 r22

Now e du 1J1= du dz ex 1 o dy du 1 2 du dy du dea dw حماد -1 o dx dy JZ 1 ex O 11 tex. o 1 2 ex oo G[R3> R₂ + Rp Volume (s)=SSS SŠ S T dudu do Hence, 0 1 1 u= exty - Now IJI 4ex gluto) utw 2 - utw=2e2 Hence, 4 e 3 Volume (3): SS S zutus dud v de o 1 1 4 3 I 4 3 2 du dudo let du dw utw & SS 5 utw) du 2

4 11 1 2 [log(utw)], dwat Tlog (3+w) – 1948) dw 4 I f log (3+w) dw -Ilog (1+2) dw 2 1 I I :- I aing (344) aw - by (6) w/ " - fo dw 3+w 4 [Int by parts] 3 = 4 log 7- 3+ w Jaw O = 4 log7 [w-3 log (3+6)] = 41og7 -[4-31og7 +3log 3] - Flog7 310g 3 -4 similarly, I:- log (It w) do log (tw).lo = 4 logo =4log5 {w-log (htw] -S ماه Itw It w ows = 41095 -[4-1095 tlog 1] -[4-log5 tlog 1] = 5log 5*log1-4 = slogs -4

Hence, Volume = s 2 [11097-3 logg- + - Slug 514] ₂ [71og7 -310g3-5logs] Ans.

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