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Summary Volume | Calculus II Notes CA$12.76
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Summary Volume | Calculus II Notes
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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.

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  • July 16, 2023
  • 4
  • 2022/2023
  • Summary

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1.6 Volume


/) * The volume

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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