5.2

Definite Integrals

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

Example Using NINT

4. Evaluate numerically.

2

1 sin

-dxxx

2 ,1 , ,sinNINT xxx 04.2

Pg. 251, 5.2 #1-35 odd