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Quick Review
42
1
4
1
3 3 3
42
1
Evaluate the sum.
1.
2. 3 1
Write the sum in sigma notation.
3. 2 3 4 ... 49 50
4. 2 4 6 8 ... 98 100
5. 3(1) 3(2) ... 3(100)
6. Write the expression as a single sum in sigma notation
n
k
n
n
k
n
4
1
0
0
3
7. Find 1 if is odd.
8. Find 1 if is even.
n
n k
k
n k
k
n
n
n
Quick Review Solutions
50
4
2
50
1
1003
1
2
1
4
1
3 3 3
Evaluate the sum.
1.
2. 3 1
Write the sum in sigma notation.
3. 2 3 4 ... 49 50
4. 2 4 6 8 ... 98 100
5. 3(1) 3(2) ... 3(100)
6. Writ
30
34
2
e the expres
3
k
k
k
n
k
n
k
k
k
k
4 42
1 1
0
0
42
1sion as a single sum in sigma notation 3
7. Find 1 if is odd.
8. Find 1 if is even.
3
0
1
nn n
n k
k
n k
k
n n
n
n n
n
What you’ll learn about Riemann Sums The Definite Integral Computing Definite Integrals on a Calculator Integrability
Essential QuestionsWhat is a definite integral and how do we find it on a calculator?
Sigma Notation
1 2 3 11
...n
k n nk
a a a a a a
The Definite Integral as a Limit of Riemann Sums
-1
0
Let be a function defined on a closed interval [ , ]. For any partition
of [ , ], let the numbers be chosen arbitrarily in the subinterval [ , ].
If there exists a number such that lim
k k k
P
f a b P
a b c x x
I 1
( )
no matter how and the 's are chosen, then is on [ , ] and
is the of over [ , ].
n
k kk
k
f c x I
P c f a b
I f a b
integrable
definite integral
The Existence of Definite IntegralsAll continuous functions are integrable. That is, if a function is
continuous on an interval [ , ], then its definite integral over
[ , ] exists.
f
a b
a b
The Definite Integral of a Continuous Function on [a,b]
1
Let be continuous on [ , ], and let [ , ] be partitioned into subintervals
of equal length ( - ) / . Then the definite integral of over [ , ] is
given by lim ( ) , where each is chon
k kn k
f a b a b n
x b a n f a b
f c x c
th
sen arbitrarily in the
subinterval.k
The Definite Integral
( )b
a f x dx
Example Using the Notation
1. The interval [-2, 4] is partitioned into n subintervals of equal length x = 6/ n. Let mk denote the midpoint of the kth subinterval. Express the limit as an integral.
n
kkk
nxmm
1
2 523lim
4
2 523 2 xx dx
Area Under a Curve (as a Definite Integral)If ( ) is nonnegative and integrable over a closed interval [ , ],
then the area under the curve ( ) from to is the
, ( ) .b
a
y f x a b
y f x a b
A f x dx
integral
of from to f a b
2. Use the graph of the integrand and areas to evaluate the integral.
3
3
29 . dxxa
232
1 Area
2
9
3
329 x dx
Area Under a Curve (as a Definite Integral)2. Use the graph of the integrand and areas to evaluate the integral.
1
1 2 . dxxb
212
1bbhArea
3 1
1 x2 dx
12
1 12
2
3
2
32
The Integral of a ConstantIf ( ) , where is a constant, on the interval [ , ], then
( ) ( ) b b
a a
f x c c a b
f x dx cdx c b a
3. Evaluate the integral .1603
0 dx
160 03 480