413
1. If , where is a continuous function, find .
2. Find the minimum value of the area of the region under the curve from to
, for all .
3. If is a differentiable function such that is never and for all , find .
;
4. (a) Graph several members of the family of functions for and look
at the regions enclosed by these curves and the -axis. Make a conjecture about how the areas
of these regions are related.
(b) Prove your conjecture in part (a).
(c) Take another look at the graphs in part (a) and use them to sketch the curve traced out by the
vertices (highest points) of the family of functions. Can you guess what kind of curve this is?
(d) Find an equation of the curve you sketched in part (c).
5. If , where , find .
6. If , find .
7. Evaluate .
8. The figure shows two regions in the first quadrant: is the area under the curve
from to , and is the area of the triangle with vertices , , and . Find .
9. Find the interval for which the value of the integral is a maximum.
10. Use an integral to estimate the sum .
11. (a) Evaluate , where is a positive integer.
(b) Evaluate , where and are real numbers with .
12. Find .
13. Suppose the coefficients of the cubic polynomial satisfy the equation
Show that the equation has a root between 0 and 1. Can you generalize this result for an
-degree polynomial?
14. A circular disk of radius is used in an evaporator and is rotated in a vertical plane. If it is to be
partially submerged in the liquid so as to maximize the exposed wetted area of the disk, show that
the center of the disk should be positioned at a height above the surface of the liquid.
15. Prove that if is continuous, then .
16. The figure shows a region consisting of all points inside a square that are closer to the center than
to the sides of the square. Find the area of the region.
17. Evaluate .
18. For any number , we let be the smaller of the two numbers and . Then
we define . Find the maximum and minimum values of if .!2 ' c ' 2t"c#t"c# !
x
1
0
f
c
"x# dx
"x ! c ! 2#
2
"x ! c#
2
f
c
"x#c
lim
nl +
'
1
s
n
s
n " 1
"
1
s
n
s
n " 2
" - - - "
1
s
n
s
n " n
(
y
x
0
f"u#"x ! u# du !
y
x
0
'y
u
0
f"t# dt
(
duf
r!
s
1 "
&
2
r
nth
P"x# ! 0
a "
b
2
"
c
3
"
d
4
! 0
P"x# ! a " bx " cx
2
" dx
3
d
2
dx
2
y
x
0
'y
sin t
1
s
1 " u
4
du
(
dt
0 ' a
*
bba
x
b
a
.x/ dx
n
x
n
0
.x/ dx
,
10000
i!1
s
i
x
b
a
"2 " x ! x
2
# dx$a, b%
lim
t
l
0
"
A"t#!B"t#"t, 0#POB"t#t0
y ! sin"x
2
#A"t#
lim
x
l
0
1
x
y
x
0
"1 ! tan 2t#
1!t
dt
f#"x#f"x# !
x
x
0
x
2
sin"t
2
# dt
f#"
&
!2#t"x# !
y
cosx
0
$1 " sin"t
2
#% dtf"x# !
y
t"x#
0
1
s
1 " t
3
dt
x
c $ 0f"x# ! "2cx ! x
2
#!c
3
fxx
x
0
f"t# dt ! $ f"x#%
2
0f"x#f
a $ 0x ! a " 1.5
x ! ay ! x " 1!x
f"4#fx sin
&
x !
y
x
2
0
f"t# dt
PROBLEMS
P R O B L E M S P L U S
O
y
xt
y=sin{≈}
A(t)
P
{
t,sin(t@)
}
O
y
xt
B(t)
P
{
t,sin(t@)
}
F I G UR E F O R P R O BL E M 8
F I G UR E F O R P R O BL E M 1 6
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