by appropriate substitutions, to
integral x 9, where R is an interval (and then evaluate
Wie of Fubini's theorem and the fundamental theorem of calculus). This
the role of the change of variables formula of the next section.
Exercises
functions. Define h: ***** by Mix,y) - $(x)by), and prove that
be , and : * -- integrable
S...--(_)(5.C)
u
SA (D.D.S - DDN-0.
Define f:Ix1**,1- {0, 1) by
Conclude as a corollary that (A x B) - 1)(B).
Use Fubini's theorem to give an easy proof that aflex èy - fy ex if these second
derivatives are both continuous. Hint: If D.D.-D.D.f>O at some point, then there
is a rectangle R on which it is positive. However use Fubini's theorem to calculate
-
f(x,y) - 1
if either x or y is irrational,
Then show that
if x and y are rational and y-ple with p and relatively prime.
(a) ... -0,
(0) ["fex, y) dy = 0 for al xe [0, 1].
f(x,y) dx = 0 if y is irrational, but does not exist if y is rational.
44 Let T be the solid torus in obtained by revolving the circle (y – a) + ' $64, in the
yz-plane, about the z-axis. Use Cavalieri's principle to compute (T) = 2m-ab?.
45 Let S be the intersection of the cylinders x + 51 and y? +2° 51. Use Cavalieri's
principle to compute (S)-1
The area of the ellipse xº/a? + yº/b> S1, with semiaxes a and b, is A = ab. Use this
fact and Cavalieri's principle to show that the volume enclosed by the ellipsoid
tě
Meten
Chand
244
IV Multiple In
we sh
factor
is v - frabe. Hint: What is the area (1) of the ellipse of intersection of the plane
*- and the ellipsoid ? What are the semiaxes of this ellipse?
4.7 Use the formula V-for, for the volume of a 3-dimensional ball of radius , and Cava.
lieri's principle, to show that the volume of the 4-dimensional unit ball BCP
**/2. Hint: What is the volume A() of the 3-dimensional ball in which the hyperplane
X-1 intersects B*?
4.8 Let C be the 4-dimensional "solid cone" in st" that is bounded above by the 3-dimer
sional ball of radius a that is centered at (0, 0, 0, A) in the hyperplane X4 -h, and below
by the conical "surface" x (x + x4 + x2). Show that (C)- mal. Note that
this is one-fourth times the height of the cone C times the volume of its base.
to an
7
mag
nei
T
OL
5 CHANGE OF VARIABLES
The student has undoubtedly seen change of variables formulas such as
[F(x, y) dx dy = ffrire
cos , r sin )r dr do
and
SMS F(x, y, z) dx dy dz
I (p sin o cos 0, p sin o sin 0, p cos p)p2 sin o dp do do,
6
f
a
b
سوم مهر »
F: RR 85 ) norm gor
ACR , BC RhA and is are contended f is integrable.
Fubinis Theoret-
href = b[la FC795) d5") *** IF 023) J**] d5*
Recally
Casx {(x) & y s fex)
f(x,y): R²R
F(x,y) dy
dy Joe
px [lem da
fi RtR ACR" is contented
CRIER, RA, 92) syag (X)]
42
S . J. dz?
A 9,2
EX c = {TOR: x + y tantha
[Cardroot) R→R
مدتها من
و [
بله (ورت = عمر
Find sf
ns2
1-,
[F(x,x) = f(x,y) dx dx
4 c. к+4+,
Ex7° ҳти №
& Freinen
f(xix, it) dx dxd
amennyi
NE
Iso kuva
.
{schot s", "Socie
C. ********
projection of the pla
X+ y +1 onto
xx-plane
fexit at holder et de se
the projection of the hyper plane X t + x + x = 1
on to xxx-space is x + x + = 1
draglustu
usin
des Gesudk
I.
up...nl
X
Y
change of Variables (substitation).
fiRtR s fex) dx toglut f fegung" (w du
J (sinki'uo causa di
lesinal" casude
F:
RR
Steel dx i= kus Raya Sf lacus) I g (2) du
glu) is non matrix of partial derivatives
9'() duo
(وند) - (a) و
a 19-92
almaq.-14)
Igū) - )
olu,..,un)
(%, ...)
is called the Jacobian
ces
Scho la calle
xy SS flocoso, e sine) F] dr do
*(*):1-(0)
SS ferry) dx dy agersino
(x,y) (ro)
(1) U (0)
KO
yorsing
9-rose 3 올
: Six
(9/9 ) > (y)
orsing
SVO)
alriol
or(sin otostolar
sino roso
K srce
tolall
f (rcos
yar side
(hyi) (1.0,0)
= 3
x = p sino Cose
y = Sinº sine
spherical coordinates
cosa
D/412)
allige
(8)
vopsino
0 %
z
3
269,49,7%a) per
sino Cose
و
sin sine
ودع
Transpir
- P sing sino
C
psino caso
من و ددي م
f caso sino
-f sino
م
i sind
SSS Scaryiz, dx dy dz.
-
()
اول مره ای بی شمار دهم / ۵ نک ne مر د ده کم یع) مررر -
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