Volume
**Volumes with Known Cross Sections
Definition: Volume of a Solid
The volume of a solid of known integrable cross section area A{x)
from x = a to x = b is the integral of A from a to b,
Example 1:
A pyramid 3 m high has congruent triangular sides and a square base that is 3 m on each side. Each
cross section of the pyramid parallel to the base is a square. Find the volume of the pyramid.
Typical
cros~
Area = base +he'ght
~
I
'A
3
x (m)
J
Fdx
o
Example 2:
A region between the graph of [ex) = 2 + x cos x and the x-axis over the interval
[-2,2] is revolved about the x-axis to generate a solid. Find the volume of the solid.
"* ractn)s
r -= 1 +)( cos (x)
curve
cros~ - section IS
a circle
A ~ lTr Z
1
J
(2
-z "IT
f XCDS
I~
(x)fdx
-me
IS
1. Find the volume of the solid whose base is a triangle bounded by y
= -2x + 2, x = 0,
and y
the
curv~t.
Examples:
= 0, and whose
cross sections are squares which are perpendicular to the x-axis.
V=f: Adx
V = S>;dx=
A-=- b-.kh \ /
-2~t2
x
V·
J.' (. 2x t2f dl< ~
~~
2. Set up (but do not integrate) integrals for the volumes of the solids with the same base as in Example 1, but whose
cross sections are:
a.
Semicircles perpendicular to the x-axis. b. Rectangles of height.!. which are
4
perpendicular to the y-axis.
d :. - 2x t 2 ~ r-- -x
. 1­ I
A"'"
A: b th
rrrZ
-Zxt2
2
I
-
I
~
f(.xtlj1dx
o
3.
Set up (but do not integrate) an integral for the volume of a solid whose base is
bounded by
i i
y = _x 2 + 2 and y = x and whose cross sections are squares
perpendicular to the x-axis.
l
11
~ J(. 2x q)d)(
o
4. Set up integrals for the volumes of the solids whose base
is the circle x
2
+ y2
=1
and whose cross sections are:
a. Equilateral triangles perpendicular to the y-axis.
b. Rectangles whose heights are three times their bases -2
and whose bases are perpendicular to the y-axis. x = -± ~- I-y l
rlShi - lefi
I_., ~" (lHJ'dy
b)
A=b+h
T L-3*2~
2 41-':1 2
I
12
I
JHr-~r d~
-I
**Disks (PerpenDisclar) (Revolve a curve around a given axis, and there is no space between the curve and the axis). Disk Formula:
v = 7r
i
b
rZ (dx or dy)
Examples: Find the volume of the solid generated by revolving the region bounded by the lines & curves about the x­
axis.
1.
Y
=x 2 ,
Y
= 0,
x= 2
2.
Y=
V9 -
x Z,
Y
=0
3
11
I (~z)2 dx
-3
3.
The region in the first quadrant bounded above by the line y
= 2,
below by the curve
and on the left by the y-axis, about the line y = 2.
y
= 2 sin x,
1-' "PPrr
O:s;
IT
x :s; 2"
curVe - lower
C\A(ve
r-: 2- 2sinx
: Z( i - S /nx)
Til]
2IT
J
o
(I - 5 I nX)l
d)(
"Washers
Washer Formula:
L
b
V = rr
(R 2 - r2) (dx or dy) R
= Outer Radius (from the axis of revolution) r
=
-
Up Pe r
- lovver
Inner Radius (from the axis of revolution)
Curve (/ongei)
cu rye (short r
(Revolve a curve around a given axis, and there is space between the curve and the axis).
Example:
Find the volume of the solid generate by revolving the region bounded by the parabolas y
= x 2 and y2 = 8x
about the x-axis.
V=A , Jr
= Tt{~ - "y)h
.r
\
TT
f (.fsX)l- (XI)2 dx
o
Example: Find the volume of the solid generated by revolving the region bounded by the lines & curves about the x-axis.
,
y
= x,
y
= 1,
x
=0
I
n S(1)2 - (xjZdx
o
Example : Set up integrals for the volumes of the solids formed by revolving the region bounded by
= _X2 + X
y
and y
a. about y
=0
=-1
b. about y
r -lower
,*~pp
~
=2
--------0-.
x
1(" _' Xl
1-)( -(-I)
(; Q-(-I)
I
IT] hZtx tl)2_ (,)2dx
o
Example: Set up integrals for the volumes of the solids formed by revolving the region bounded by y
y=x+2
a.
b.
about the x-axis
about the line y
=4
x1-:.)(tZ
X2 _ ~
-2-= 0 -t--~---i+-'-~x (X -2) ( Xt I) 0 !:
-\
x~2,-1
R;
R: L\-XZ
X t 2-0
(: 4-' (X t 2)
r:X.lo
l
Tf
J IXtZ)1
-I
-
(>(1)2dx
1.
IT
I r-(1-x _2)1 dx
(1-x Z
-I
=x 2
and
"Shells (ParaShell)
http://www.youtube.com/watch?v=V6nTsxumjgU&feature=relmfu&safety mode=true&persist safety mode=l&safe=a
ctive
v = Lb 2rr(shell radius)(shell hight)
-the.19ht
Example:
+he
IS
c.urve
y
o
Examples: Use the cylindrical shell method to find the volume of the solid.
1.
Y
= x,
Y
= I,
x
=0
2. Y
= 2x -
1,
Y
= {X,
x
=0
-[X ~ Zx-I
X ~ ~X2 -~xtl
~~2 5x
(~X
X~
r ~ )<
h~
-D - (lx-I)
LIT JX (~ -2~ tl) ax
o
~
0
-I)(x - ()
t \
1ft X" I