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WORKSHEET 32  Fall 1995
1. Find all continuous functions
f
(
x
) satisfying
Z
x
0
f
(
t
)
dt
=[
f
(
x
)]
2
+
C
(Hint: Di±erentiate both sides with respect to
x
.)
2. Let
A
betheaverageva
lueo
f
f
(
x
)=
x
3
on the interval [1
,
3]. Determine
A
, then graph
y
=
f
(
x
)and
y
=
A
on the interval [1
,
3] on the same graph.
3. Suppose
f
(
x
)=
1
2
√
x
+1
. Evaluate the following de²nite integrals:
a)
Z
3
0
f
0
(
x
)
dx
b)
Z
3
0
f
(
x
)
dx
c)
Z
3
0
± Z
x
0
f
0
(
t
)
dt
²
dx
4. A parabola opens downward, and has roots at 0 and
a
(here
a>
0).
a) Suppose the area enclosed by the parabola’s arch and the
x
axis is 1. What is the highest point of
the parabola in terms of
a
? First draw a picture and make a guess. Then compute an answer.
b) Suppose the maximum height height of the parabola is 1. What is the area enclosed by the
parabola’s arch and the
x
axis in terms of
a
? Again, ²rst make a guess.
5.
a) Find the area of the ²nite plane region bounded by the curve
y
=
x
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This note was uploaded on 04/11/2011 for the course MATH 1400 taught by Professor Grether during the Spring '08 term at North Texas.
 Spring '08
 Grether

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