Introduction to integrals Integral, like limit and derivative, is another important

concept in calculus Integral is the inverse of differentiation in some sense There is a connection between integral calculus and

differentiation calculus. The area and distance problems are two typical

applications to introduce the definite integrals

The area problem Problem: find the area of the region S with curved sides, w

hich is bounded by x-axis, x=a, x=b and the curve y=f(x). Idea: first, divide the region S into n subregions by partitio

ning [a,b] into n subintervals [xi-1,xi] (i=1,,n) with x0=a and xn=b; then, approximate each subregion Si by a rectangle since f(x) does not change much and can be treated as a constant in each subinterval [xi-1,xi], that is, Si¼(xi-xi-1)f(i), where i is any point in [xi-1,xi]; last, make sum and take limit if the limit exists, then

the region has area

1n

ii

S1

lim ,

n

in

i

S

1

lim .

n

in

i

S

Remark In the above limit expression, there are two places of signif

icant randomness compared to the normal limit expression: the first is that the nodal points {xi} are arbitrarily chosen, the second is that the sample points {i} are arbitrarily taken too.

means, no matter how {xi} and {i} are

chosen, the limit always exists and has same value.

1

lim

n

in

i

S S

1

lim

n

in

i

S

The distance problem Problem: find the distance traveled by an object during the

time period [a,b], given the velocity function v=v(t). Idea: first, divide the time interval [a,b] into n subintervals;

then, approximate the distance di in each subinterval [ti-1,ti] by di¼(ti-ti-1)v(i) since v(t) does not vary too much and can be treated as a constant; last, make sum and take

limit if the limit exists, then the distance in the

time interval [a,b] is

1n

ii

d

1

lim ,

n

in

i

d

1

lim .

n

in

i

d d

Definition of definite integral We call a partition of the

interval [a,b]. is called the size of the partition,

where are called

sample points. is called Riemann sum. Definition Suppose f is defined on [a,b]. If there exists a

constant I such that for any partition p and any sample points

the Riemann sum has limit then we call

f is integrable on [a,b] and I is the definite integral of f

from a to b, which is denoted by

0 1 1: n np a x x x x b

1max{ }

ii n

x

1( 1, , ). i i ix x x i n 1[ , ]( 1, , ) i i ix x i n

1

( )

n

i ii

f x

i0

1

lim ( ) ,

n

i ii

f x I

( ) .b

aI f x dx

Remark The usual way of partition is the equally-spaced partition

so the size of partition is

In this case is equivalent to

Furthermore, the sample points are usually chosen by

or thus the Riemann sum is given by

, 0,1, ; ( ) /ix a ih i n h b a n

( ) /h b a n

1i ix

0 n

i ix

1

0 1

( ) ( )( ) or ( )

n n

i i

b a i b a b a i b af a f a

n n n n

Example Ex. Determine a region whose area is equal to the given

limit

(1) (2)10

1

2 2lim (5 )

n

ni

i

n n

1

lim tan4 4

n

ni

i

n n

Definition of definite integral In the notation a and b are called the limits of i

ntegration; a is the lower limit and b is the upper limit; f(x) is called the integrand.

The definite integral is a number; it does not depend on x, that is, we can use any letter in place of x:

Ex. Use the definition of definite integral to prove that

is integrable on [a,b], and find ( ) f x c

( ) ,b

af x dx

( )b

af x dx

( ) ( ) ( ) . b b b

a a af x dx f t dt f r dr

.b

acdx

Interpretation of definite integral If the integral is the area under the

curve y=f(x) from a to b If f takes on both positive and negative values, then the int

egral is the net area, that is, the algebraic sum of areas

The distance traveled by an object with velocity v=v(t), during the time period [a,b], is

( )b

af x dx( ) 0,f x

( )b

af x dx

v( ) .b

at dt

Example Ex. Find by definition of definite integral. Sol. To evaluate the definite integral, we partition [0,1] int

o n equally spaced subintervals with the nodal points

Then take as the sample points. By taking limit to the Riemann sum, we have

1 2

0x dx

/ , 0,1, , . ix i n i n / i ix i n

1 2 230

1 1

1 ( 1)(2 1) 1lim ( ) lim ( ) lim .

6 3

n n

i in n n

i i

i n n nx dx f x

n n n

Example Ex. Express the limit into a

definite integral. Sol. Since we have

with Therefore,

The other solution is

1 1 1lim( )

1 2 2

n n n n

1 1 1,

1

in i nn

1 1

1 1 1 1 1 1( ),

1 2 2 1

n n

i i

if

in n n n n nn1

( ) .1

f xx

1

0

1 1 1 1lim( ) .

1 2 2 1

n

dxn n n x

2

1

1. dx

x

Example Ex. If find the limit

Sol.

1 2 ( 1)lim (sin sin sin ).

n

n

n n n n

0

sin 2, xdx

0

1 2 ( 1)lim (sin sin sin )

1 2lim (sin sin sin )

1 2sin .

n

n

n

n n n nn

n n n n

xdx

Exercise1. Express the limits into definite integrals:

(1)

(2)

2. If find

1 2 ( 1)2 2 21

lim (1 ).n

n n n

ne e e

n

2 2 2 2 2 2

2 2 2lim( ).

4 1 4 2 4n n n n n

2sin sin sin

lim( ).1 112

n

nn n n

n n nn

0sin 2, xdx