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Volume
, aloowk
1. Fin he volume of Me old that 16 obotalned — by revolved te region 1
We X-ome
Raition «
(xi 42,
An Wis protien, we wil vce fhe Dick Method, From Ae giaph OF Y * Mt
witica
Me Wat Ae wads y Ne repre idk 6 yy che Wwht OF *
Teckangle . Therefore te Volume Of he ctuk i6
DY = wy*hx
envitiog Ye (xia pint, FR hove x ab 9. Sbetng Te
en how
value we will
AV = TC (xt 2x 43)'4x
y= TW (xt-ax* Hox* ~ 12x44) 4x
Summing HE Volumes of He aids aed taiag the Kits We fled stat He Volane a
He solid 6
3
VS J (0-48 Hox? “ne 44) ax
Ver feats SEs Px dy.
Tt ee = ax" oe ~ tt +a.
Ve Si yt go. a+ %
ee xt ray
s 0
Yer [xo -cott Bat wert 99] 2. Find the volume of he volid Yat {¢ obtained by volving the tegion —aleout
te ne y= S.
Soliton :
q jeal rectangle is tevolved about i 26
fa represents vertical gle. oes about the line Y 7 > the
Traub —golid ic a washer with Outer acting g- x? ier radius
4 —X 5 and widh dx. Therefore its volume ic
- S23\i- (8 -y\°
WV - w [(E-v) -(§ xy | bx
AV = 1 (x" ~&x?- x" 45x) by
Summing he volumes Of “he Washers Ord baking the limit, We find thot the
Rqyied volume ix
Ve fx (x52 -¢ + 5x) Ax suded 09 We Gags
3. Find the volume of ead Generated fay revolving fhe region Mave
AF NE Kquotions and ineqpaliiuc about Me y-axis
Neyr ele, X20, yet yyed
AV 71x? Ay !
= (Viurys) 4 ;
> yflorye
AV = 1 (wiy*)ay ha a ‘ "y
TE we um the volume of hese clicks Gocd Yoke du lity we find Prot the required
Volume is
+
Ve rf loryt dy
ae,
2 [ Ny aby ]
aM * [hw ~[ucds L Lil]
(
See | Ww
tS + 3) i ion boundud
1 Fe he wane of Re ig geneva Wy rerling “he region lou
BS Ye gophc eg Phe equations —alpaut Ane indicated avis.
BER Sy ro, xd) Kes 5 he X-Ois
We tate and inner radius othe washer Qtnerated bythe teprecentottin vertical
Matangle ak f(x) = x ond GX) = 0 ,respectiey. Therefore, ite volume ig
BV = EAT ay
AV = TOR) ax
String we volumes of A Waites ord faking he tity Fe find Pat
heqyived volume {¢
$
Veal ae
3
s
a= |
S s
