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5.3 Volumes of Solids of Revolution

Disk Method

Standards :

MCI 1

MCI 1c

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Old Area between curves

Let's consider flxkx '- 4×+10 & glxl = 4x - x ! Find the area of reglin

between * 1 & x =3.

n

f↳=×2jx +10

|$nmDmDMnBH

#~tg**iiµ±4×+103 - GX - xD dx

= { xt -4×+10 - txtxhdx

= } 2×2-8×+10 dx

= 2¥ . 8×21+10 × = 25×3 - 4×2+10 ×} =

= Gm'- 1435+10(D - [ It 's

'- 445+1011D=

=@

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2nd Practice problem : fkt . De 0 ex I 2,

find the area.

t.EE#H*!EH.onx=y÷ : }xnI= Ehiy . Glos"D

= It . } of

= 2¥ = 189

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n@Vo1umesofSolidsofRerolutimlDiskMethod1Now.letsconsidertheprevionsexamplewherenmwewillfindthe.V0lume_3DM0DEl.I

.

FnmTop€

: :

. ÷4 . : .

Toe'

.

'

tenth'i⇒HI E÷fI:"a "

Ii⇐⇒%¥jltx

. axis'

¥

: ''

B - y :#

indignantEt@IY0Y.a, .flthxrtdx :p Txdx £

=*Ixdx=#x÷ }Itfyrtzarde

Gma

=TFx 't Volume of Circle

=t[kD . [Hey =Ara*dx

=@=TFxY2d×

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So let's generalize! Let's consider a curve y= fed & find the volume

revolving around x - axis .

^ y- axis

ii

÷iµE÷*× ±Hi÷HBi.i ! "

4, ,

Ittx

. axisIfaseaimfµ§IT . axis

3dFm&i

A= ITR = THKH'

V= A * dx= tfflxlpdx

' ' Take the sum of ALL OF THE DISKS "

ume=§*H×n↳ - Froefmoiilnagwrhoennwd

x - axis

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Example Bflndthevdumeofthesolidby revolving around x - axis ofy=x2,

y=O and ×=2 .

*

's Vi "fstr2dx= §MxYdx

÷

= pTx4dx=t§×4dx

1iiili'*¥t⇒t¥,

i3}@

. "%