Integrals
15.6 Surface
1123
be an lil!F143' Investigatian
E : .ri + yi + 2zk
i lectrical Charge Let
(a) Use a computer algebra system to graph the vector-valued
!'itrostatic field. Use Gauss's Law to find the total charge
-:losed by the closed surface consisting of the hemisphere
function
= JT -7=V
and its circular base in the -rry-plane.
(4 - v sin a) sin(2u)j +
{u,v) : (4 u sin u) cos(2u)i +
l
<
v
S
l'
v c o s a k 0, < u < t ,
-.nt of Inertia In Exercises33 and 34, use the following
,-rulas for the momentsof inertia about the coordinateaxes
, .urface lamina of densitYP.
= I ltr'+z')p(x,v,z)ds
= I ltr'+zz)p(x,t,z)ds
- | ltr'+y2)p(x,y,z)ds
_'sJ
' erify that the moment of inertia of a conical shell of uniform
:ensity about its axis is lma2, where n is the mass and a is the
',dius and height.
,
r erify that the moment of inertia of a spherical shell of uniform
.:ensiiyabout its diameter is lmaT, where m is the mass and a is
:re radius.
-
nent of Inertia In Exercises 35 and 36, find I" for the given
-rina
with uniform density of 1. Use a computer algebra
-rem to verifY Your results.
This surface is called a Mtibius strip.
(b) Explain why this surface is not orientable'
(c) Use a computer algebra system to graph the space curve
representedby r(u, 0). Identify the curve'
(d) Construct a M<ibius strip by cutting a strip of paper'
making a single twist, and pasting the ends together'
(e) Cut the Mtjbius strip along the space curve graphed in part
(c), and describe the result.
44. Consider the vector field
F(x,y, z) : zi + ri + Yk
and the orientable surface S given in parametric form by
- v)i + u2k,
r(a, v) : (u + vz)i + (u
O S u < 2 , - 1 < v = 7 .
(a) Find and interPret r! x rv.
(b) Find F ' (r, x r,) as a function of u and v'
: , : r - 1 y 2 : A 2 , 0 < Z = h
(c) Find u and v at the point P(3, 1.4).
*
(d) Explain how to find the normal component of F to the
surface at P. Then find this value.
. = x 2 + y 2 , 0 < z 3 h
'r' Rate In Exercises 37 and 38, use a computer algebra
.rem to find the rate of mass flow of a fluid of density p
'rrugh the surface S oriented upward if the velocity field is
: . en by F(t,y,z) = 0.52k.
-
(e) Evaluatethe flux integral
J /r
' N as'
5:z= 16-x2-y2, z>0
. . S :z : J T 6 = 7 = T
of OneSheet
Hyperboloid
19. Define a surface integral of the scalar function / over a
Explain how to evaluate the surface
surface z = g6,i.
integral.
Consider the parametric surface given by the function
ll). Describe an orientable surface.
(a) Use a graphing utility to graph r for various values of the
constants a and b. Describe the effect of the constants on the
shapeof the surface.
(b) Show that the surface is a hyperboloid of one sheet given by
{1. Define a flux integral and explain how it is evaluated'
{2. Is the surface shown in the figure orientable? Explain'
r(u,v): acoshacosvi* acoshusinvj * bsinhzk'
x2
--|'
a'
-y;2- - - z 2 - , r '
o'
a'
(c) For fixed values u + tts, describe the curves given by
r(ao, v) : a cosh ao cos vi * a cosh ao sin vj + b sinh uok'
(d) For fixed values v : v0, describe the curves given by
Double twist
r ( a , v o ):
a c o s h u c o s v o i+ a c o s h u s i n v o j * D s i n h r ' r k '
: (0' 0)'
(e) Find a normal vector to the surface at (u, v)
1078
C h a p t el r5
V e c t oA
r nalysis
F o r c u r v e s r e p r e s e n t e d b y -Sv@
: ) , a< x < b , y o u c a n l e t x : / a n d o h u , :
parametric form
x:t
and y:g(t), a<t<b.
Becausedx : dt for this form, you have the option of evaluatingthe line intc::the variable.r or the variabler. This is demonstratedin Example 9.
Form
in Ditferential
a LineIntegral
EXAMPLE
E Evaluating
Evaluate
C:y=4x-x2
I
I ydx+x2dy
JC
where C is the parabolicarc given by y : 4x - xz from (4,0) to (1,3), as sh, F i g u r e1 5 . 1 8 .
Solution Rather than converting to the parameter I, you can simply ret,::
variable x and write
dy:(4-2x)dx.
!:4x-x2
Then, in the direction from (4,0) to (1,3), the line integralis
f
f\ _ x2\dx + x2\4- 2rl dx)
I t,.t* * x2d1,: | [t+"
J+
Jc'
:
fl
J,
r+r * 3/ - 2x])dr
t
- ; lxalr : ; 69
l2x2+ r'
z
zl1
I
\ccFrrrnrplcT.
Finding Latersl Surface Area The figure below shows a piece of tin that i:'
been cut from a circular cylinder. The baseof the circular cylinder is modek':
by x2 + y2 : 9. At any point (;, ,y)on the base,the height of the object is
given by
J'k.9:
I*cos
o.
Explain how to use a line integral to find the surface area of the piece of rin
-,a.r,
\1-r,y)
12.1
.r. 75-80, determine the interval(s) on which the
irtd function is continuous.
76.
vrr -f vr - l.l
i-arcsintj+(t-l)k
,'i*e /j+ln(r l)k
'. Ir. tant)
80. "(t) =
(s."a,ra)
3I + 1k.
.r \ector-valuedfunction s(t)
:'nration
of r.
841
In Exercises 89 and 90, two particles travel along the space
curves r(r) and u(t). A collision will occur at the point of
intersection P if both particles are at P at the same time. Do the
particles collide? Do their paths intersect?
89. r(0 - rzi + (9r - 20[ + r]k
u(0 : (3r + 4)i + rrj + (sr - 4)k
9 0 . r ( r )- d + / r j + r - r k
u ( / ) : ( - 2 r + 3 ) i + 8 r j + ( 1 2 1 +2 ) k
Thittk About It In Exercises 91 and 92, two particles travel
along the space curves r(t) and u(t).
:!'r the vector-valuedfunction
:ri + (r
V e c t o r - V a l uF
eu
dn c t i o n s
91. If r(t) and u(r) intersect,will the particlescollide']
is the specified
r'rticaltranslationthree units upward
,)rizontaltranslationtwo units in the directionof the
-.rllVe -r-axlS
:,,rizontaltranslation
five r"rnits
in the directionof the
. rtir e r'-axis
hc definition of continuity of a vector-valued
r. Give an exampleof a vector-valuedfunction that
:d but not continuousaI t : 2.
:tlge of a playgroundslide is in the shapeof a helix of
rlcters.The slide has a height of 2 rnetersand rnakes
lL-tcrevolution fiom top to bottorn. Find a vector;tion fbr the helix. Use a computeralgebrasystemto
. function.(Thereare many correct
answers.)
92. If the particlescollide, do their pathsr(r) and u(r) inrersect?
True or False? In Exercises 93-96, determine whether the
statement is true or false. If it is false, explain why or give an
example that shows it is false.
93. If i g. and /r are first-degreepolynornial functions, then the
curvegivenby .r : ./(t), .r' - g(l). and : : i(t) is a line.
94. If the curvegiven by r : l(t). J - g(/). and : : l(r) is a line.
then.l, g, and /i are first-degreepolynomial functions of r.
95. Two particlestravel along the spacecurves r(r) and u(r). The
intersectionof their pathsdependsonly on the curvesh'acedout
by r(r) andu(t), while collisiondependson rheparamererizations.
9 6 . T h e v e c t o r - v a l u e df u n c t i o n r ( t ) :
lies on the paraboloid.r: r'r + ir.
rri + rsinrj + /coslk
Witchof Agnesi
t the fbllowing vector-valuedfunctions represent
graph'l
. ( 3 c o s r * l ) i + ( 5 s i n r+ 2 ) j + 4 k
- - + i+ ( - 3 c o s 1 * l [ + ( 5 s i n r + 2 ) k
- ( 3 c o s r- l ) i + ( - 5 s i n t - 2 ) : + 4 k
- ( - 3 c o s2 r + l ) i + ( 5 s i n 2 t + 2 ) j + 4 k
In Section3.5, you studieda famous curve called the Witch of
Agnesi.In this projectyou will takea closerlook at this tunction.
Considera circleofladius a centeredon ther'-axisat (0, a). Let
A be a point on the horizontalline r' : 2a. letO be the origin, and
let B be the point where the segmentOA intersectsthe circle. A
point P is on the Witch of Agnesi if P lies on the horizontalline
throughB and on the vertical line throu_sh
A.
(a) Show that the point A is tracedout by the vector-valuedfunction
.i u(l) be vector-valued
functionswhoselimits exist
'r'rrte that
uri)l- lEr(r)xlhu(r).
.: u(t) be vector-valuedfunctions whose lirnits exist
': rrre that
u r)l -
lim r(d . lh u(d.
i r is a vector-valuedfunction that is continuousat
:. contlnuousat a'.
:hL'converseof Exercise87 is not true by finding a
:J tirnctionr suchthat llrll is continuousat c'but r
I il()l-ls at c .
r.(9) : 2acot0i t 2ttj, 0 < 0 < tr
where 0 is the angle Ihat OA rr-rakes
with the positive.r'-axis.
(b) Show that the point B is tracedout by the vector-valuedfunction
r u @ )- a s i n 2 I i + u ( l - c o s 2 0 ) . i . 0 < 0 < r .
(c) Combine the resultsof parts (a) and (b) to frnd the vectorvalued function r(0) fbr the Witch of Agnesi. Use a glaphing
utility to glaph this curve fbr a - l.
( d ) Describethe lirnits lirr r(0) and lim r(9).
d-{,
u -n
( e ) Eliminate the parameter 0 and deterrnine the rectangular
equationof the Witch of Agnesi.Use a graphingutility to graph
this function for o : I and compare your graph with that
obtainedin part (c).