Question

(13 pts) Use a triple integral to find the volume of the given solid. The solid within the cylinder x2 + y2 = 9 and between the planes 2 = 1 and x + 2 = 5.

Answer 1

Asolars criven the solid within the cylinder se?ty2mg and between the plane z=1 and atz=5

20 10 -5 5

Here, limit of Z is 2=1 to 2-5-2 We have to find volume of solid region. رقاN / s where, dr=dardy de Volume & volume 221 Isl. dv JUL Izdy dre volume = J J [2], dy dan // [(5x) – 1] dy R 7 volume dydn volume = JJ (4-2) dy du R since, R: a²ty² = 9 put x=rcoso y=rsino 13,09=25 Then, Ircoso ² + (8 sino) 2=g 38² (Cos204 8220) = 9 82=9 coro +8020 = 1 positive 8=13 since, radies is Then, 8=3 so, limit of or is and limit of o is to 223 to o=22 O=0

Also replace dy dre by J1 dido where, du Da ә0 gy ४ ay Caso -rsind & J Ing oloso & J a scoso +8 kmo >) Il >r (20152201) le replace dyde by sdo do Whing above results and limits in we have voheme- (4 - 82080) rdo do 02 s20 2 volume 2 & volume = 9 Volume - La (40-820049) do do pa (22 - Caso I do {"(2x02_ox cosa ) -(0-0) de a) volume pr 118 -9 coso) do 0 – 9698 (18x27 – 9 cm(2x)) – Crexo-gam(o))) 2 volume 2 [(36A - 9x) -(0-0)] Volume 2 >) [180 rolume 2 {(1sx volume 2 volume. 36h 0 367/ u

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