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1VIATH132O: Volumes by Cylindrical Shells
Instructor: Laura Strube
31 August 2015
Turn in for participation credit
So(u+iOflc
Name and Class ID:
1. Set up and evaluate the integral for the volmue of the solid that results when the region
0 is rotated about the line x 4. Begin
bounded by y =
+ 2x + 3, y = 0, and x
by sketching the region.
(a) Complete the square to convert y
=
2 + 2x + 3 into the form: y
—x
=
a(x
—
2 + A:.
li)
(i (i)aQ
2
—(x
(b) Sketch the region and then sketch the solid.
4
4%
1
k
‘C
4.
d smrncfr
(c) Sketch an approximating shell on the solid.
1.x
1 s&--c c on -tue
(d) For this problem, is the width of the shell L x or
‘V
S& sk-eh CLôt.,€:
(i.e. we
1
(e) Is the height of the shell measured alollg the x or y axis? ‘Write the equation for
the height of the shell.
Se
£ & f-h
-
aLo v e: h ih
j:
is
iettu rd
7J
0Jo
24243
pejh+=
(_X2#243
—,
(°
3
_X22X+
(f) Is the radius of the shell measured along the x or y axis? Write the equation for
the circumference of the shell.
J-tdiLcs IS ,Te
LL’t1.S aJ.Onff
1
A
:
—
x)
7me
x ‘s
x
X
W
C a, 33
J // J1Q 4t& I
(ts she/I zx izzdis 4—
S. fl7tZLLL4
7C
(g) Write the equation for the volume of a single shell.
srj.e she/I:
Su-rIu tzrz
/Oli%rn-e ôí
2ff(4-x(
XZ+X3)
s;nJecJr/L
(h) Write the integral for the volume of the solid and solve.
3
VoIu-rr
=r
[Your solution should be H]
(4_x(xa÷ +3)cJx