MATH132O:
Arc
Volumes,
Length,
and
Mean
Value
of
a
Function
lustructor: Laura Strube
02 September 2015
Turn in for participation credit
Ors
Name and Class ID:
1. i\’Iean Value Theorem for Integrals
Given f(r) =
:r, find all the values c in the interval [—3,
—
51
such that f(c)
f.
5
;COVf
S-1:
=
-
F, 12S
ç
-
-J
I
Set
cv
,
-
/
Ni
G( N
-
I to
5
soke4
o
2
r
3-3Y-i(o -O
qI2
+
L
+
—
(9
zC3
3]
2
X3
-
2. Mean Value Theorem for Integrals: Application
Suppose you are driving on a straight highway on which the speed limit is 55 mu/hr. At
8:05pm a police car clocks your velocity at 50 mi/hr and at 8:10 ani a second police car,
posted 5 mi down the road, clocks your velocity at 55 mi/hr. Explain why the police
have the i’ight to charge you with a speeding violation.
(Hint: Velocity is change in position over time. i.e. the velocity function is the derivative
of the position function)
)frfl9:
• -J-irm
V/o
=
50
2h
(poi+ior)
(veLci4-)
-
h rS
h rs
Em ‘n
-
hr
•
Th
5
V()5
-L€XI
ye
‘liz
I
(
v(
c/
-
)7
Ir?rr
_I_-7
7_
iJ
ç psthn
meni
r: foc’
Av
5 mTh
j) rn
)
/2
v€/’
&
1L4
l
wc4-iLr
jitLfl ze
h’cô po) tt.
‘o ô mph.
in
•
Is
Th Th
(; e.
tC
,fl
1
9
LJ
s
chr
2
(
po
t
1po
ifl
fl..o.S
3. Arc Length
Find the arc length of the curve y = x
2 from (1, 1) to (2, 2v’) by integrating appro
/
3
priately with respect to p.
(Hint: After setting up the arc length integral, factor y
3 out of both of the terms in
/
2
the square root function)
Ar-c r1i rmMJ
(t’jçJ
rc2Ld
b
L
312
x=f
z
-3
f2/3a
-
2(2/3)
-r
1
iJ
J-.
2
1A1
z
2-
-“3d
.3
Iz
-
=
—
(3I
3f-7
(Z!
=
I\jT,
z
L_1
---.—
t:
X)
4. Volume by Washers
Find the volume of the solid obtained by rotating the region bounded by y
Sr about the line r = 4. Begin by sketching the solid.
isc(t-c-inee(
=
,2
and
irt;-h-e
mrrti
C
4,
ichcc
-
her
Fh-ci. c-t
-
Ai-c-i n fltr
=
-
Ove,I
V o twn-e
4iLS(,SL
X
CA%<
=
spec
-a
J7N
VcLn o
be sohd
f2Z4
Y1d
(4
j(g_]d
-
4
_---Fi--
“L
3
=
G+3)22j
0
‘1ie
1)oShe
n [(4-4 (i