MATH 2414 notes and examples

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61 Volumes




Volume Tr h




Let Plx Plane perpendicular to the X axis
and passing thru thepoint


Area of the intersection of
P x with S
A x



1 DIVIDE a b IN SUBINTERVALS of EQUAL
WIDTH DX WITH ENDPOINTS a Xo X Xny x b


2 CUT S INTO n SLABS BY SLICING S ALONG
THE PLANE Pix

, 3 VOLUME OF 1 SLAB




f dl
DX AXT
h




reima
xi
iÉ DX

Xi t x 5 v him
Dx Alxt

Alx dx




EXAMPLES

1 FIND THE VOLUME OF THE SOLID OBTAINED BY
non BOUNDED by e
y 0 x
1511111 E


4 8



YIFI
re

A x Tr
1 2
Tile


V0 YE e'd T ex tile

, 126 1
VOLUME The dx T ex




2 FIND THE VOLUME OF THE SOLID OBTAINED BY ROTATING THE
REGION BOUNDED BYTHE CURVES y X3 y x 0 ABOUTTHE
x axis

y

Alx TLV TL V2
TL x TL

F x2 x
o




VOLUME A x dx


Tex
6 dx


It

, ROTATING AROUND
Y AXIS
y
Aly dy

91g
Aly




EXAMPLES

FIND THE VOLUME OF THE SOLID GENERATED BY ROTATING
THEREIGON BOUND BY THE CURVES Y X2 X 0 y 4
ABOUT THE Y AXIS




Aly Tly 2



y v TC To
y y dy St



o


VOLUME Aly dy