Thomas' Calculus, Media Upgrade (11th Edition)

Review Problems for Exam 4
(Covers Chapter 5)
1.
( A calculator can be used for this problem) Use finite approximation to estimate
the area under the graphs of the function using
i) A lower sum with four rectangles of equal width.
ii) An upper sum with four rectangles of equal width.
iii) The midpoint rule (rectangles whose height is given by the value of the
function at the midpoint of the rectangle’s base) with four rectangles of equal
width.
a) . 1 0 ) ( 3 = = = x and x between x x f
b)
. 5 1
1
) ( = = = x and x between
x
x f
2. Express the sums in sigma notation.
a)
1 + 4 + 9 + 16
b) 2 + 4 + 6 + 8 + 10
c)
6
1
5
1
4
1
3
1
2
1
1 - + - + -
3.
Suppose that
5
1
=
∑
=
n
k
k
a
and
7
1
=
∑
=
n
k
k
b
. Find the values of
a)
∑
=
n
k
k
a
1
3
b)
∑
=
+
n
k
k
a
1
) 1 (
c)
∑
=
-
n
k
k k
b a
1
) 2 (
4.
Suppose that
∫ ∫ ∫
= = - =
5
1
5
1
2
1
. 8 ) ( , 6 ) ( , 4 ) ( dx x g dx x f dx x f
Find
a)
∫
1
5
) ( dx x f
b)
∫
5
5
) ( dx x f
c)
∫
5
2
) ( dx x f
d)
∫
-
1
5
)] ( 2 ) ( 3 [ dx x g x f
e)
∫
-
2
5
) ( 2 dx x f
5. Find the average value of the function over the given interval.