Less 22 G

16.5.49 14.7.63 61,53

GFI f Hajj dzdydx

2 Fye ayn Z 7

IY f f fo r i rdzdrd0

O

165.492

Region bdd by f 4cosy f 2 zzo

2

i

revolved about the t axes

NZ

PKHHqfiucaiefsnfnue.MIL'Enotawathe volume corresponding to this

MDMy

region

NZ

Intersection occurs when2 p 4 cosy 4 Iz

y

So 0 E I E Is and p ranges from the

orange sphere to the blue sphere that is

2E PE 4Cosa

And OE OE 2T This gives

1251 124054 p2sincedpd4do

222T3

Now the volume of the whole blue sphereis V f it 2

3 32 53

so the volume of the gray region is

3232 2232T 1105J

1 61 Vol of sphere f 9

Spherical

fo fo p since dpd4d0O

or 8 f f Jo p Sind dpdado

cylindrical

Iz 9 get8 fo fo fo r dzdrdo

usprojectto Xy plane

redmmkgrgg.gs igtTdzdydx