Explains how to set up an integral to solve for volume for general shapes and circular symmetry. Provides formulas and examples for solids of revolution; disk method, washer method, and shell method.
the
neight sel
of this shape would be cross sectional A multiplied by
A Tr2
·
11//////)
=
V Tr
= 2 n
For
"
general shape:"volumes by slicing
i
I >X
X*
approximate the area of each slice by a cylinder of volume A(X) AX
add the areas and take a limitas n -> x(Ax -
0)
v lim" A(X,*)AX Sa
= =
A(x) dX
n 00k 1
- =
solid has semicircular base of radius 5 and cross sections perpendicular
e.g. a a to
the base and parrallel to the diameter are squares:
yz)
=
4(25
e 2 y 25
A(y) 2
=
x
"
-
+ =
x 1125
=
-
yz
I
S
>X
l = 25 -
y2 25
+ -
y2
229
yz
= -
v SA(y)dy S-4(25 yz)dy
=
=
-
special Cases -
Circular
symmetry
"Solids ofRevolution"
eg0 y xx
on Co, 27
revolving aboutthe x axis
=
In
A(X) Tr π(X)2
ex
=
=
>r xX
=
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