¥4,4 't - pitt.edukaveh/lecture-math0230-sep25-2017.pdframanujan found very nice approx. for...
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sep.es/2o=f#jpp@pTomorrow 's quiz : Sec
. 7.2 & 7.3
( volumes )
Are length :
¥4,4't" §'t#d×
\ length of
the vec .
diff
(X, fcx ) ) - ( 1,
f{ x ) ) ( 1,
flex) )
-
Ex. ( # 17 see . 7. 4)
y=ln ( l - x2 ) of xszt
¥the are length : fcx , = ln ( l -
XZ )
setup an integral ffx ) = ¥2 . ( - zx ) .
that computes ...
kef 'tF⇐Ta× #
E_×( Circumference of an ellipse )
a¥tb¥= 1
#( x. YM )
•#•nWrite an integral that gives Circum .of the
a ellipse .ofitchdx×=
- a Alternatively , You
Y = b2-a×÷b2#- can use implicit
diff .
1
y '= - . -2×(6%2) .
2¥21a2
Effie = zfaa FEEL ax
41( bt k¥2)
( You Can little bit simplify this expression ) .
.It took centuries to finally prove
that
above integral can not be expressed in
terms of ' ' elementary function"
e- g . sin , cos
& polynomials . # ppgexpiln
. Ramanujan found very nice approx . for this
integral( Movie : The man who knew infinity ) .
÷skip . 7.5 - surface area .
÷-
Applications
in physics .
General Principle :
X, Y two quantities
Third
gudgiven by XY C product ) .
( when X & Y Constant ) .
§pieexmvle : M¥7 FEELEY
Now suppose X is variable & Y depends on )/
( i.e. Y=fc × ) ) .
Then :b
The third quantity = ) fcx ) dxfor values of ×Idb ×=a
It Word W=Force×distanmultiplication Formula
If
FF Constant
=D then W=F . d.
d
•when F is variable Say Fix ) , × distance to 0 .
Fcx ) ( or position )
→¥-0a x b
.
Work done by force Fcx )
w@DIn.x" to '' ⇒ is :
↳ sumEx.int. Work - Work by force of gravity
a by force of a Spring
tmno. pressure ( hydrostatic pressure )
.Center of mass &momentum
rest
¥ Hook 's law Mdt-
#×
F =-K@ µooh is const
b
Work done to
stfoetnhxatmosrxinsb=f- kx dx
( const . K known )X=a
.
Next time :
work needed to empty the pool ?
¥F⇒-
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