MATH 1431
EXAM 3 REVIEW
Upper sums, lower sums, Riemann sums
Let
f
(
x
) =
x
2

4
x
+ 6
on
[

1
,
4],
and let
P
=
{
1
,
0
,
1
,
2
,
3
,
4
}
be a partition of
[

1
,
4].
1
1
2
3
4
x
2
4
6
8
10
y
1. Calculate the lower sum
L
f
(
P
)
and the upper sum
U
f
(
P
).
2. Let
s
1
, s
2
, s
3
, s
4
be the midpoints o f the subintervals determined by
P
. Use these values to calculate
the corresponding Riemann sum
S
(
P
).
3. Calculate
4
0
f
(
x
)
dx
. Which of the values in (a) and (b) gives the best approximation of the definite
integral?
4. Calculate the average value of
f
on the interval
[0
,
4].
The Fundamental Theorem of Calculus; Properties of the Definite Integral
1. Show that
F
(
x
) =
x
√
16 +
x
2
is an antiderivative for
f
(
x
) =
16
(16 +
x
2
)
3
/
2
. Use this result to evaluate
3
0
16
(16 +
x
2
)
3
/
2
dx
.
2. The function
G
is defined by:
G
(
x
) =
cos
x
1
t
1

t
2
dt
.
(
i
)
Determine
G
(2
π
)
(
ii
)
Determine
G
(
π/
6)
3. Evaluate the definite integral
1
0
5
x
(1 +
x
2
)
4
dx
.
4. Assume that
f
is a continuous function and that
2
0
f
(
x
)
dx
= 3
,
3
0
f
(
x
)
dx
= 1
,
5
3
f
(
x
)
dx
= 8
.
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 Spring '08
 Any
 Calculus, Riemann Sums, dx, 1 g, Riemann

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