Ma 113 Functions And Inverse Functions, The Exponential Function And The Logarithm Worksheet Page 33
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41Worksheet # 25: Definite Integrals
1
2
1
2
1. Suppose
( )
= 2
( )
= 3
( )
=
1 and
( )
= 4.
0
1
0
0
Compute the following using the properties of definite integrals:
2
2
0
(a)
( )
(d)
( )
+
( )
1
1
2
2
2
1
(b)
[2 ( )
3 ( )]
(e)
( )
+
( )
0
0
2
1
(c)
( )
1
2. Suppose that
is a continuous function and 6
( )
7 for
in the interval [3 9].
9
(a) Find the largest and smallest possible values for
( )
.
3
4
(b) Find the largest and smallest possible values for
( )
.
8
x
3. Suppose that we know
( )
= sin( ), find the following integrals.
0
π
(a)
( )
0
π
(b)
( )
π/2
π
(c)
( )
π
5
3
if
3
4. Find
( )
where ( ) =
if
3
0
5. Recognize the sum as a Riemann sum and express the limit as an integral.
n
3
lim
4
n
i=1
6. Let ( ) =
and consider the partition
=
which divides the interval [1 3] into
0
1
n
subintervals of equal length.
(a) Find a formula for
in terms of
and .
k
(b) We form a rectangle whose width is (
) and whose height is (
). Give the area of the
k
k 1
k
rectangle.
(c) Choose the sample points to be the right endpoint of each subinterval and thus
=
.
1
2
n
Form the Riemann sum
(
) and use the formulae for sums of powers to simplify the Rie-
mann sum.
(d) Take the limit as
tends to infinity to find the area of the region
= (
) : 1
3 0
.
(e) Find the area of
using geometry to check your answer.
7. Simplify
b
c
a
( )
+
( )
+
( )
a
b
c
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