Calculus
Name___________________________________ ID: 1
©d h2R0J1H5u MKUuKt_aC JS_o^f]tPwDa[rBel pL\LECX.R O dA\l\lE Er\iDgjhktosV xr_eosgeJrVvgeDd_.
Assignment
Date________________ Period____
For each problem, find the volume of the solid that results when the region enclosed by the curves is revolved about the the given axis. 3
1) y = 1, y = x , x = 0 Axis: y = 0
2) x = - y 2 + 4, x = 0 Axis: x = 0
For each problem, find the volume of the solid that results when the region enclosed by the curves is revolved about the the x-axis. 3) y =
x + 3, y = x 2 + 3
4) y = 1, y =
3
x, x = 0
For each problem, find the volume of the solid that results when the region enclosed by the curves is revolved about the the given axis. 5) y = x 2 + 2, y = 1, x = -2, x = 0 Axis: y = -2
6) y = - x 2 + 7, y = - x + 5, x = 0, x = 1 Axis: y = 2
Worksheet by Kuta Software LLC
-1-
©R U2P0T1g5H KK[uQtuaL pSHoMfktRw_acrfeq HLNLaCN.m ` EAklfla trPiBgkhZtysT jrcefsdeArPvxeJdd.m Y fMJaNdHeE [wIiwtYhM aIHncfQi`nbiQtpep ACFaXlLceuAlbu]sh.
For each problem, use the method of cylindrical shells to find the volume of the solid that results when the region enclosed by the curves is revolved about the the y-axis. 7) y = 2 x + 1, y = x 2 + 1
1
8) y = 5, y = , x = 2 x
For each problem, use the method of cylindrical shells to find the volume of the solid that results when the region enclosed by the curves is revolved about the the given axis. 9) y = 2 x + 1, y = x 2 + 1 Axis: x = -3
10) y = x + 3, y = x 2 + 3 Axis: x = -3
For each problem, find the area of the region enclosed by the curves. 11) y = -
2
2,
y = -2,
x x = -4, x = -1
12) x = y , x = 2 y , y = 0, y = 4
Worksheet by Kuta Software LLC
-2-
©r _2z0Y1e5K cKCu\tvau `S_oafHtMwiakroeG NLtLwCx.b h RAGlMlZ SruiugJhBt`s_ crneYs_eDrgv^e\d`.O ^ WM[atdKez Tw^iBtGhr eIEn_fnicnoi^tKe\ JCqaBldcBuQlJuosX.
13) y = 2 x 3 + 3 x 2 - 4 x, y = x 2
14) y =
x3
2
- 3 x, y =
x2
2
Worksheet by Kuta Software LLC
-3-
©r T2P0s1b5b pKWuAtLav sSRoCfottwNaQr[eu lLRLMCi.^ g bAplClP Wr`iCgLhptjsL mrxeHsOexrMviendj.e W IMqaadZeM DwsiftZhH LISnefOiCnNiPteed DC\aZlKcru`lPu`sP.
Calculus
Name___________________________________ ID: 1
©^ G2U0d1_5D `KcuWtVaX CSBo\fctMwwanrieu VLOLkCN.P F NAGlwlP qr]iggxhitPsa zrvebsZecrrvxendG.
Assignment
Date________________ Period____
For each problem, find the volume of the solid that results when the region enclosed by the curves is revolved about the the given axis. 3
2) x = - y 2 + 4, x = 0 Axis: x = 0
1) y = 1, y = x , x = 0 Axis: y = 0 1
p
ò
2
(1 - ( 3 x )2) dx
ò
(- y 2 + 4) 2 dy
=
512 p » 107.233 15
p
0
-2
2 5
= p » 1.257
For each problem, find the volume of the solid that results when the region enclosed by the curves is revolved about the the x-axis. x + 3, y = x 2 + 3
3) y = 1
p
ò
((
4) y = 1, y = 1
)
2
x + 3) - ( x + 3) dx 2
2
p
x, x = 0
(1 - ( 3 x ) 2) dx
0
0
2 5
23 p » 7.226 10
=
ò
3
= p » 1.257
For each problem, find the volume of the solid that results when the region enclosed by the curves is revolved about the the given axis. 5) y = x 2 + 2, y = 1, x = -2, x = 0 Axis: y = -2 0
p
ò
6) y = - x 2 + 7, y = - x + 5, x = 0, x = 1 Axis: y = 2 1
((x2 + 4)2 - 3 2) dx
p
-2
=
ò
((-x2 + 5)2 - (-x + 3)2) dx
0
626 p » 131.109 15
=
233 p » 48.799 15
Worksheet by Kuta Software LLC
-1-
©` I2c0T1w5] iKJuttsak \S[oOfNt\wRanrBet kLVLaCZ.D o FAFlhlG LrrivgVhptjsN mrveXsIelrivieRdE.c t bMSaSdreZ ^wlittFhs wIUnRfoijnQiptLeB TC_aRlTcwumlguTsq.
For each problem, use the method of cylindrical shells to find the volume of the solid that results when the region enclosed by the curves is revolved about the the y-axis. 7) y = 2 x + 1, y = x 2 + 1 2
1
8) y = 5, y = , x = 2 x
ò
2p x(2 x + 1 - ( x 2 + 1)) dx
2
ò( )
2p x 5 -
0
8 = p 3
1 5
=
1 x
dx
81 p 5
For each problem, use the method of cylindrical shells to find the volume of the solid that results when the region enclosed by the curves is revolved about the the given axis. 9) y = 2 x + 1, y = x 2 + 1 Axis: x = -3
10) y = x + 3, y = x 2 + 3 Axis: x = -3
2
1
2p ( x + 3)( x + 3 - ( x 2 + 3)) dx
ò
0
ò
32 = p 3
23 = p 10
2p ( x + 3)(2 x + 1 - ( x 2 + 1)) dx
0
For each problem, find the area of the region enclosed by the curves. 11) y = -
2
2,
y = -2,
x x = -4, x = -1 -1
ò ( - + 2) -4
=
2
x2
12) x = y , x = 2 y , y = 0, y = 4 4
ò (2
dx
y-
y ) dy
0
=
9 = 4.5 2
16 » 5.333 3
Worksheet by Kuta Software LLC
-2-
©g O2I0u1v5b NKvuxtSao YSEosfQtzwJaQrLec yLzL\Cs.W M WAFljlS frJiPgFhQtRsV jrZeWs^efrSvpe_dc.l E lMmaYdQeo [w_iQt`hH vICnafSiRnmiUtXec UCIaOlZc`umlKuxsI.
13) y = 2 x 3 + 3 x 2 - 4 x, y = x 2
14) y =
0
ò (2 + 2 ò ( - (2 x
3
x - 4 x) dx + 2
0
2
x2
- 3 x, y =
2
ò (2 2) ò ( 2 - ( 2 - 3 ))
-2 1
x2
x3
-2 3
x 3 + 3 x 2 - 4 x)) dx
0
37 = » 6.167 6
x3
x2
- 3x -
x2
x3
dx +
x dx
0
=
253 » 10.542 24
Worksheet by Kuta Software LLC
-3-
©N L2p0b1q5k dK_ulttaT bScoofetvwUaerueo WLqLSCg.Z f CAJlolD mrYiOgMhPtQsR nrseFskeyrxvqe[dq.A Z BM_awdzea HwaiMtnhx ]ImnQfUilnJibteeT lCeailPcIujlCutsA.
