Question

LE 6) he can find the volume of the above slid by Goshe ruled the convien keskussaith Small Rickmel tile face of being solutit oly ke X-a . We get an crasher (The outure of he washers than in Hack) The code d' h has along Va+1, invol has readwi, We will indigd te werk. He wahe sking the length of the solid. Clearly its coill give the area of Me mol Honce. Ito a widomy lecian. C At the tickness of our warhan is da The area of the surface of the weather in A s T ( + 1² m . The volume of the washer as Adx [(+9*-nl]da | An carter can be imagined as a cylnou taken out I from a larger cylinda] Answer to d) and e):- d Nano to fad the volume we integrate the above rolume Ada varying a le from 0 to 4. 5o the volume of the sold v=shade Scra+0²-1)da ("11 (2+2/2+ i) - ) da (nt 2+2) da *), e da 1 20/*<<da .+ 11162 - 7.40 1929. 561 The release to 56 (e) The solid in the question entirely contained in the aylinders were crued solara bunded by robotny tre 409. Doight for le 2: 4. ye volume legyenchen 10 **3*4 = 36 .-- The solid entirely contains the smalle oplonder. whose curved surface in formed by pelag o 20 on bot, 24 on noi This cylinder has volume TTX144.40 (2) led with baking (out

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Answer 1

7 by we can sind the volume of the above solid by washer method, we consider the strip with small thickness on the surface of drevolution of rotate it along the x-aros. We get a washes ( The outside of the washer to is shown in black). The autside of washer has radius inside pin has .. sadius we will integrate the volume of the washer along the length of the solid.clearly this will give the area of the solid . Hence its a valid cross section. ) Let the thickness of our washer is da The area of the surface of the washer is A = T (Tetoj" - 11") The volume of the washes is Adx = [1(77 +1) ~_ ] ax [An washer can be imagined as a cylinder taken out from a larges cylinder / Answer to d) and e) :- d) Thus to find the volume we integrate the above volene Ads varying n ie from 0 to 4

of the solid So the toon of volume the volume ton 2 - Jade The smal is f - ringa +1) * - rl) dx aj (11 (2+257 +1) - n) or - 1 (ap 217 ) ar The » 200 + 200 stie ax / f 211 x 32 1 2:17 16 X 2 2 17.8 T 1742 2 :"( 24432) 5611 The volume is 56 e) The solid in the question is entirely contained in the cylindes where curved surface is bounded by no tating the line f(x) =3 ... right to red left to x=4. The roleome of this cylinder is

TIX 3 y 4 = 36 11 11) The solid entirely contains the stor smaller cylinders whose curved surface is formed by sotating thx) = 1 to x=o on left x = 4 an right The cylinder has volume nxiux 4 = 417 12) ta