Math 152
Workshop 1
Spring 2011
1. Suppose that
f
is a continuous function (deﬁned for all
x
) and that it is known that
Z
1
0
f
(
x
)
dx
= 5
,
Z
1

1
f
(
x
)
dx
= 3
,
Z
2
0
f
(
x
)
dx
= 8
and
Z
4
0
f
(
x
)
dx
= 11
.
Evaluate the integrals:
(a)
Z
2
0
f
(2
x
)
dx
(b)
Z
π
0
(sin
x
)
f
(cos
x
)
dx
(c)
Z
3
2
xf
(8

x
2
)
dx
.
(
Suggestion:
Use substitutions, such as
u
= cos
x
in (b).)
2. Sketch the region
R
deﬁned by 0
≤
y
≤
1
/x
3
, 1
≤
x
≤
2.
(a) Find (exactly) the number
a
such that the line
x
=
a
divides
R
into two parts of equal
area.
(b) Then ﬁnd (to 3 places) the number
b
such that the line
y
=
b
divides
R
into two parts
of equal area.
3. Which has more area, the region in the ﬁrst quadrant enclosed by the line
x
+
y
= 1 and the
circle
x
2
+
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 Spring '11
 SOSA
 Calculus, Derivative, Integrals, dx

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